Bevezet\303\251s a matematik\303\241ba J\303\241rai Antal Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak.
<Text-field style="Heading 1" layout="Heading 1">1. Halmazok</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">2. Term\303\251szetes sz\303\241mok</Font></Text-field>
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">3. A sz\303\241mfogalom b\305\221v\303\255t\303\251se</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">4. V\303\251ges halmazok</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">5. V\303\251gtelen halmazok</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">6. Sz\303\241melm\303\251let</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">7. Gr\303\241felm\303\251let</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">8. Algebra</Text-field>
<Text-field style="Heading 2" layout="Heading 2">8.1. Csoportok</Text-field> restart;with(group); Nz5JKURlcml2ZWRTRzYiSSRMQ1NHRiRJLk5vcm1hbENsb3N1cmVHRiRJLFJhbmRFbGVtZW50R0YkSS1TbkNvbmp1Z2F0ZXNHRiRJJlN5bG93R0YkSS1hcmVjb25qdWdhdGVHRiRJJ2NlbnRlckdGJEksY2VudHJhbGl6ZXJHRiRJJWNvcmVHRiRJJ2Nvc2V0c0dGJEknY29zcmVwR0YkSShkZXJpdmVkR0YkSSllbGVtZW50c0dGJEksZ3JvdXBtZW1iZXJHRiRJK2dyb3Vwb3JkZXJHRiRJJmludGVyR0YkSShpbnZwZXJtR0YkSSppc2FiZWxpYW5HRiRJKWlzbm9ybWFsR0YkSStpc3N1Ymdyb3VwR0YkSSltdWxwZXJtc0dGJEkrbm9ybWFsaXplckdGJEkmb3JiaXRHRiRJJ3Bhcml0eUdGJEkocGVybXJlcEdGJEklcHJlc0dGJEkrdHJhbnNncm91cEdGJA==
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.1. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.2. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.3. Homomorfizmusok.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.4. P\303\251lda.</Font></Text-field> a^(x+y);expand(%); KUkiYUc2IiwmSSJ4R0YkIiIiSSJ5R0YkRic= KiYpSSJhRzYiSSJ4R0YlIiIiKUYkSSJ5R0YlRic= 1^2; (-1)^2; 0+0; IiIi IiIi IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.5. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.6. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.7. Reprezent<Font encoding="UTF-8">\303\241</Font>ci<Font encoding="UTF-8">\303\263</Font>k.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.8. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.9. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.10. P\303\251lda.</Font></Text-field> solve(x+x=0); solve(x*x=1) assuming real; IiIh NiQiIiIhIiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.11. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.12. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.13. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.1.14. <Font encoding="UTF-8">K\303\266vetkezm\303\251ny: egyszer\305\261s\303\255t\303\251si szab\303\241ly.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.15. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.16. P\303\251ld\303\241k.</Font></Text-field> solve(z^6=1,z); G:=evalc([%]); expand(evalc(1/G[3]));expand(evalc(G[2]*G[3])); NighIiIiIiIsJComI0YkIiIjRiQpLCZGKEYjKiYqJkYoRiReI0YkRiRGJCkiIiRGJ0YkRiRGJ0YkRiMsJEYmRiQsJComRidGJCksJkYoRiMqJiwkRixGJEYkRi5GJEYjRidGJEYjLCRGMkYk NyghIiIiIiIsJiNGJCIiI0YjKiYsJComRiZGJF4jRiRGJEYkRiQpIiIkRiZGJEYjLCZGJkYkKiZGKkYkRixGJEYkLCZGJkYjRi9GJCwmRiZGJEYoRiM= LCYjIiIiIiIjISIiKiYqJkYjRiReI0YkRiRGJCkiIiRGI0YkRiQ= LCYjIiIiIiIjISIiKiYsJComRiNGJF4jRiRGJEYkRiQpIiIkRiNGJEYm undefine('`&*`'); define('`&*`','multilinear','flat','identity'=1); &*(i,i):=-1;&*(j,j):=-1;&*(k,k):=-1;&*(i,j):=k;&*(j,k):=i;&*(k,i):=j;&*(j,i):=-k;&*(k,j):=-i;&*(i,k):=-j; (-1*i)&*(-1*k)=i&*k; ISIi ISIi ISIi SSJrRzYi SSJpRzYi SSJqRzYi LCRJImtHNiIhIiI= LCRJImlHNiIhIiI= LCRJImpHNiIhIiI= LywkSSJqRzYiISIiRiM= undefine('`&*`'); define('`&*`','flat','orderless','identity'=e); &*(a,b):=c;&*(a,c):=b;&*(b,c):=a;&*(a,a):=e;&*(b,b):=e;&*(c,c):=e; e&*a&*c&*b; SSJjRzYi SSJiRzYi SSJhRzYi SSJlRzYi SSJlRzYi SSJlRzYi SSJlRzYi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.17. Geometriai p\303\251ld\303\241k.</Font></Text-field> D3:=grelgroup({tau,epsilon},{[epsilon,epsilon,epsilon],[tau,tau],[epsilon,tau,epsilon,tau]}); LUkqZ3JlbGdyb3VwR0koX3N5c2xpYkc2IjYkPCRJJHRhdUdGJUkoZXBzaWxvbkdGJTwlNyVGKUYpRik3JEYoRig3JkYpRihGKUYo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.19. Feladat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.20. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.22. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.23. <Font encoding="UTF-8">R\303\251szf\303\251</Font>lcsoport, r<Font encoding="UTF-8">\303\251szcsoport.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.25. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.27. \303\201ll\303\255t\303\241s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.28. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.29. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.30. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.31. Gener\303\241tum.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.1.32. P\303\251ld\303\241k: line\303\241</Font>ris transzform<Font encoding="UTF-8">\303\241</Font>ci<Font encoding="UTF-8">\303\263</Font>k csoportjai.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.33. \303\201ll\303\255t\303\241s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.34. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.1.35. Rend.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.36. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.37. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.38. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.39. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.40. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.41. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.42. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.43. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.44. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.45. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.46. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.47. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.48. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.49. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.50. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.51. Mell\303\251koszt\303\241lyok.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.52. Lagrange t\303\251tele.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.53. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.54. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.55. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.56. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.57. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.58. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.59. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.60. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.61. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.62. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.63. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.64. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.65. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.66. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.67. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.68. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.69. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.70. Norm\303\241loszt\303\263.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.71. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.72. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.73. P\303\251lda.</Font></Text-field> H1:=subgrel({x=[tau]},D3); isnormal(H1); H2:=subgrel({x=[epsilon]},D3); isnormal(H2); LUkoc3ViZ3JlbEdJKF9zeXNsaWJHNiI2JDwjL0kieEdGJTcjSSR0YXVHRiUtSSpncmVsZ3JvdXBHRiQ2JDwkRitJKGVwc2lsb25HRiU8JTclRjBGMEYwNyRGK0YrNyZGMEYrRjBGKw== SSZmYWxzZUclKnByb3RlY3RlZEc= LUkoc3ViZ3JlbEdJKF9zeXNsaWJHNiI2JDwjL0kieEdGJTcjSShlcHNpbG9uR0YlLUkqZ3JlbGdyb3VwR0YkNiQ8JEkkdGF1R0YlRis8JTclRitGK0YrNyRGMEYwNyZGK0YwRitGMA== SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.74. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.75. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.76. Bels\305\221 automorfizmusok.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.1.77. Centraliz\303\241tor \303\251</Font>s centrum.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.1.78. Oszt\303\241lyegyenlet.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.79. P<Font encoding="UTF-8">\303\251ld\303\241k.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.80. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.81. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.1.82. Faktorcsoport.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.83. P\303\251ld\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.1.84. Homomorfizmus magja.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.85. Homomorfizmust\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.86. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.87. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.88. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.89. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.90. Projekt<Font encoding="UTF-8">\303\255</Font>v csoportok.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.91. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.92. Direkt szorzat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.93. V\303\251ges Abel-csoportok alapt\303\251tele.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.1.94. V\303\251gesen gener\303\241lt Abel-csoportok alapt\303\251tele.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.95. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.96. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.97. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.98. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.99. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.100. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.101. Feladat: diszkr<Font encoding="UTF-8">\303\251</Font>t direkt szorzat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.102. Cayley t\303\251tele.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.103. Permut\303\241ci\303\263csoportok.</Font></Text-field> convert([3,4,2,1,7,6,5],'disjcyc'); convert(%,'permlist',7); NyQ3JiIiIiIiJCIiIyIiJTckIiImIiIo NykiIiQiIiUiIiMiIiIiIigiIiciIiY= undefine('`&*`'); `&*`:=(x,y)->mulperms(y,x); g:=convert([3,4,2,1,7,6,5],'disjcyc'); h:=convert([2,5,3,4,1,7,6],'disjcyc'); f:=g&*h; convert(%,'permlist',7); Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLV9JJmdyb3VwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlSSltdWxwZXJtc0dGJTYkRiZGJEYlRiVGJQ== NyQ3JiIiIiIiJCIiIyIiJTckIiImIiIo NyQ3JSIiIiIiIyIiJjckIiInIiIo NyQ3JCIiIiIiJTcnIiIjIiIoIiInIiImIiIk NykiIiUiIigiIiMiIiIiIiQiIiYiIic= invperm(f); invperm(h)&*invperm(g); NyQ3JCIiIiIiJTcnIiIjIiIkIiImIiInIiIo NyQ3JCIiIiIiJTcnIiIjIiIkIiImIiInIiIo S5:=permgroup(5,{[[1,2]],[[1,2,3,4,5]]}); grouporder(S5); isabelian(S5); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiImPCQ3IzckIiIiIiIjNyM3J0YrRiwiIiQiIiVGJw== IiQ/Ig== SSZmYWxzZUclKnByb3RlY3RlZEc= H1:=permgroup(5,{[[1,2,3,4,5]]}); grouporder(H1); isabelian(H1); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiImPCM3IzcnIiIiIiIjIiIkIiIlRic= IiIm SSV0cnVlRyUqcHJvdGVjdGVkRw== elements(H1); PCc3IjcjNyciIiIiIiMiIiQiIiUiIiY3IzcnRiZGKkYpRihGJzcjNydGJkYpRidGKkYoNyM3J0YmRihGKkYnRik= gg:=RandElement(S5); H2:=permgroup(5,{gg}); grouporder(H2); groupmember([[1,2,3,4,5]],H2); groupmember(gg,H2); NyQ3JCIiIiIiIzckIiIkIiIl LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiImPCM3JDckIiIiIiIjNyQiIiQiIiU= IiIj SSZmYWxzZUclKnByb3RlY3RlZEc= SSV0cnVlRyUqcHJvdGVjdGVkRw== inter(H1,H2); issubgroup(H1,S5); issubgroup(H1,H2); isnormal(H1,S5); isnormal(H2,S5); isnormal(H2,H1); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiImPCI= SSV0cnVlRyUqcHJvdGVjdGVkRw== SSZmYWxzZUclKnByb3RlY3RlZEc= SSV0cnVlRyUqcHJvdGVjdGVkRw== SSV0cnVlRyUqcHJvdGVjdGVkRw== SSZmYWxzZUclKnByb3RlY3RlZEc= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.104. T\303\251tel.</Font></Text-field> parity(g); parity(h); parity(f); IiIi ISIi ISIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.105.K\303\266vetkezm\303\251ny.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.106. P\303\251lda.</Font></Text-field> [[1,3]]&*[[1,2]]&*[[3,4]]; [[1,2]]&*[[2,1]]; NyM3JiIiIiIiIyIiJCIiJQ== NyI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.1.107. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.108. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.109. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.110. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.111. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.112. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.113. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.114. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.115. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.116. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.117. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.118. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.1.119. Defin\303\255ci\303\263.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn M11:=permgroup(11,{[[1,2,3,4,5,6,7,8,9,10,11]],[[3,7,11,8],[4,10,5,6]]}); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiM2PCQ3IzctIiIiIiIjIiIkIiIlIiImIiInIiIoIiIpIiIqIiM1Ric3JDcmRi1GMUYnRjI3JkYuRjRGL0Yw L:=[1,2];L[1]:=3;L; NyQiIiIiIiM= IiIk NyQiIiQiIiM= grouporder(M11); IiU/eg== conjugateclasses:=proc(G) local GG,SS,p,i,f; SS:=[]; GG:=elements(G); for p in GG do f:=false; for i to nops(SS) do if areconjugate(G,SS[i][1],p) and not p in SS[i] then f:=true; SS[i]:=SS[i] union {p}; break; fi; od; if not f then SS:=[op(SS),{p}]; fi; od; convert(SS,set); end; Zio2I0kiR0c2IjYnSSNHR0dGJUkjU1NHRiVJInBHRiVJImlHRiVJImZHRiVGJUYlQyY+Rig3Ij5GJy1fSSZncm91cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUkpZWxlbWVudHNHRiVGIz8mRilGJ0kldHJ1ZUdGNEMlPkYrSSZmYWxzZUdGND8oRioiIiJGPS1JJW5vcHNHRjQ2I0YoRjhAJDMtX0YySS1hcmVjb25qdWdhdGVHRiU2JUYkJiZGKDYjRio2I0Y9Rik0LUkjaW5HRjQ2JEYpRkhDJT5GK0Y4PkZILUkmdW5pb25HRjQ2JEZIPCNGKVtAJDRGKz5GKDckLUkjb3BHRjRGQEZVLUkoY29udmVydEdGNDYkRihJJHNldEdGNEYlRiVGJQ== conjugateclasses(M11):nops(%); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.1.120. P\303\251lda.</Font></Text-field> S4:=permgroup(4,{[[1,2]],[[2,3]],[[3,4]]}); grouporder(S4); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiIlPCU3IzckIiIiIiIjNyM3JCIiJEYnNyM3JEYsRi8= IiND elements(S4); A4:=permgroup(4,select(x->parity(x)=1,%)); grouporder(A4); isnormal(A4,S4); cosets(S4,A4); PDo3IjcjNyQiIiIiIiM3JEYlNyQiIiQiIiU3IzckRiZGKjcjRik3IzclRiZGJ0YqNyM3JkYmRidGKkYrNyM3JEYnRio3IzcmRiZGKkYrRic3IzcmRiZGKkYnRis3IzcmRiZGK0YqRic3IzcmRiZGJ0YrRio3IzcmRiZGK0YnRio3IzclRidGKkYrNyM3JUYmRipGKzcjNyVGJkYrRic3IzclRiZGKkYnNyM3JEYnRis3IzclRiZGJ0YrNyM3JUYnRitGKjckRi1GSDcjNyVGJkYrRio3IzckRiZGKzckRlFGNA== LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiIlPC43IjckNyQiIiIiIiM3JCIiJEYnNyM3JUYsRi1GLzcjNyVGLUYvRic3IzclRixGL0YnNyM3JUYsRidGLTcjNyVGLEYvRi03IzclRixGLUYnNyM3JUYtRidGLzckNyRGLEYvNyRGLUYnNyM3JUYsRidGLzckNyRGLEYnNyRGLUYv IiM3 SSV0cnVlRyUqcHJvdGVjdGVkRw== PCQ3IjcjNyQiIiQiIiU= N1:=permgroup(4,{}); grouporder(N1); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiIlPCI= IiIi N2:=permgroup(4,{[[1,2],[3,4]],[[1,3],[2,4]],[[1,4],[2,3]]}); grouporder(N2); isnormal(N2,A4); cosets(A4,N2); LUkqcGVybWdyb3VwR0koX3N5c2xpYkc2IjYkIiIlPCU3JDckIiIiIiIjNyQiIiRGJzckNyRGK0YuNyRGLEYnNyQ3JEYrRic3JEYsRi4= IiIl SSV0cnVlRyUqcHJvdGVjdGVkRw== PCU3IjcjNyUiIiMiIiQiIiU3IzclRiZGKEYn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.1.121. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.122. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.123. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.1.124. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.125. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.126. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.127. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.128. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.129. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.130. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.131. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.132. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.1.133. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.1.134. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">8.2. Gy\305\261r\305\261k \303\251s testek</Font></Text-field> restart;
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.1. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.2. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.3. P\303\251ld\303\241k.</Font></Text-field> undefine('`&*`'); `&*`:=(x,y)->[x[1]*y[1],x[2]*y[2]]; [1,0]&*[0,1]; Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQqJiZGJDYjIiIiRi4mRiZGLUYuKiYmRiQ2IyIiI0YuJkYmRjJGLkYlRiVGJQ== NyQiIiFGIw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.4. P\303\251lda.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.5. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.6. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.7. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.8. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.9. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.10. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.11. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.12. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.13. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.14. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.15. Homomorfizmusok.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.16. P\303\251ld\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.2.17. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.18. T\303\251tel.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.19. <Font encoding="UTF-8">T\303\251tel</Font>.</Text-field> X:={0,1,2,3,4}; map(x->x+x mod 5,X); map(x->x+x+x mod 5,X); map(x->x+x+x+x mod 5,X); map(x->x+x+x+x+x mod 5,X); PCciIiEiIiIiIiMiIiQiIiU= PCciIiEiIiIiIiMiIiQiIiU= PCciIiEiIiIiIiMiIiQiIiU= PCciIiEiIiIiIiMiIiQiIiU= PCMiIiE= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.2.20. G<Font encoding="UTF-8">y\305\261r\305\261 karakterisztik\303\241</Font>ja.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.2.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.22. R\303\251szgy\305\261r\305\261, ide\303\241l.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.23. P\303\251ld\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.24. P\303\251ld\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.25. P\303\251ld\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.2.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.27. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.28. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.29. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.30. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.31. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.32. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.33. Reprezent<Font encoding="UTF-8">\303\241</Font>ci<Font encoding="UTF-8">\303\263</Font>k.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.34. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.35. Boole-gy\305\261r\305\261k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*8.2.36. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.37. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.38. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.39. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.40. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.41. Feladat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*8.2.42. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.43. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.44. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.45. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.2.46. Feladat: Stone t<Font encoding="UTF-8">\303\251tele.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.47. Mell\303\251koszt\303\241lyok.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.48. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.49. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.50. Faktorgy\305\261r\305\261.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.51. P\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.52. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.2.53. Homomorfizmus magja.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.54. Homomorfizmus-t\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.55. P\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.2.56. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.57. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.58. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.59. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.60. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.61. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.62. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.63. Direkt szorzat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.64. P\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.65. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.66. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.67. Gauss-gy\305\261r\305\261k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.68. P\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.69. Euklideszi gy\305\261r\305\261k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.70. \303\201ll\303\255t\303\241s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.71. P\303\251lda: Gauss-eg\303\251szek.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">8.2.72. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.73. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.74. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.75. B\305\221v\303\255tett euklideszi algoritmus.</Font></Text-field> polynomexgcd:=proc(a,b,z) local x0,x1,x2,y0,y1,y2,r0,r1,r2,q; x0:=1; y0:=0; r0:=a; x1:=0; y1:=1; r1:=b; do if r1=0 then return [x0,y0,r0] fi; q:=quo(r0,r1,z); r2:=expand(r0-q*r1); x2:=expand(x0-q*x1); y2:=expand(y0-q*y1); r0:=r1; x0:=x1; y0:=y1; r1:=r2; x1:=x2; y1:=y2; od; end; Zio2JUkiYUc2IkkiYkdGJUkiekdGJTYsSSN4MEdGJUkjeDFHRiVJI3gyR0YlSSN5MEdGJUkjeTFHRiVJI3kyR0YlSSNyMEdGJUkjcjFHRiVJI3IyR0YlSSJxR0YlRiVGJUMpPkYpIiIiPkYsIiIhPkYvRiQ+RipGNz5GLUY1PkYwRiY/KEYlRjVGNUYlSSV0cnVlRyUqcHJvdGVjdGVkR0MtQCQvRjBGN083JUYpRixGLz5GMi1JJHF1b0dGJTYlRi9GMEYnPkYxLUknZXhwYW5kR0Y+NiMsJkYvRjUqJkYyRjVGMEY1ISIiPkYrLUZKNiMsJkYpRjUqJkYyRjVGKkY1Rk4+Ri4tRko2IywmRixGNSomRjJGNUYtRjVGTj5GL0YwPkYpRio+RixGLT5GMEYxPkYqRis+Ri1GLkYlRiVGJQ== debug(polynomexgcd); SS1wb2x5bm9tZXhnY2RHNiI= polynomexgcd(z^3+z+1,z^2+2,z); {--> enter polynomexgcd, args = z^3+z+1, z^2+2, z IiIi IiIh LCgqJClJInpHNiIiIiQiIiJGKEYlRihGKEYo IiIh IiIi LCYqJClJInpHNiIiIiMiIiJGKEYnRig= SSJ6RzYi LCYiIiJGI0kiekc2IiEiIg== IiIi LCRJInpHNiIhIiI= LCYqJClJInpHNiIiIiMiIiJGKEYnRig= IiIh IiIi LCYiIiJGI0kiekc2IiEiIg== IiIi LCRJInpHNiIhIiI= LCZJInpHNiIhIiIiIiJGJQ== IiIk LCYiIiJGI0kiekc2IkYj LCgiIiJGIyokKUkiekc2IiIiI0YjISIiRiZGKQ== LCYiIiJGI0kiekc2IiEiIg== IiIi LCRJInpHNiIhIiI= IiIk LCYiIiJGI0kiekc2IkYj LCgiIiJGIyokKUkiekc2IiIiI0YjISIiRiZGKQ== LCYjIiIiIiIkRiQqJkYjRiRJInpHNiJGJCEiIg== IiIh LCYjIiIjIiIkIiIiKiYjRiZGJUYmKUkiekc2IkYkRiZGJg== LCgqJiMiIiIiIiRGJUkiekc2IkYlISIiRiRGKSomRiRGJSlGJ0YmRiVGKQ== IiIk LCYiIiJGI0kiekc2IkYj LCgiIiJGIyokKUkiekc2IiIiI0YjISIiRiZGKQ== IiIh LCYjIiIjIiIkIiIiKiYjRiZGJUYmKUkiekc2IkYkRiZGJg== LCgqJiMiIiIiIiRGJUkiekc2IkYlISIiRiRGKSomRiRGJSlGJ0YmRiVGKQ== <-- exit polynomexgcd (now at top level) = [1+z, 1-z^2-z, 3]} NyUsJiIiIkYkSSJ6RzYiRiQsKEYkRiQqJClGJSIiI0YkISIiRiVGKyIiJA== undebug(polynomexgcd); SS1wb2x5bm9tZXhnY2RHNiI= polynomexgcd(z^3+z+1,z^2+2,z); NyUsJiIiIkYkSSJ6RzYiRiQsKEYkRiQqJClGJSIiI0YkISIiRiVGKyIiJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.76. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.77. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.78. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*8.2.79. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.80. Maxim\303\241lis ide\303\241l.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.81. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.82. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.83. Pr\303\255mide\303\241l.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.84. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.85. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.86. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.87. H\303\241nyadostest.</Font></Text-field> r1:=(z^3-1)/(z^2-1); r1:=simplify(r1); r2:=simplify((z^4-1)/(z^3-1)); r1*r2; KiYsJiokKUkiekc2IiIiJCIiIkYpRikhIiJGKSwmKiQpRiYiIiNGKUYpRilGKkYq KiYsKCokKUkiekc2IiIiIyIiIkYpRiZGKUYpRilGKSwmRilGKUYmRikhIiI= KiYsKiokKUkiekc2IiIiJCIiIkYpKiQpRiYiIiNGKUYpRiZGKUYpRilGKSwoRipGKUYmRilGKUYpISIi KiYsJiIiIkYkSSJ6RzYiRiQhIiIsKiokKUYlIiIkRiRGJCokKUYlIiIjRiRGJEYlRiRGJEYkRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.2.88. K\303\266vetkezm\303\251ny.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.2.89. Algebrai strukt\303\272r\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.2.90. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.91. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.92. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.93. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.2.94. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.95. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.96. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.97. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.2.98. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">8.3. Polinomok</Text-field> restart;with(PolynomialTools); NzNJMENvZWZmaWNpZW50TGlzdEc2IkkyQ29lZmZpY2llbnRWZWN0b3JHRiRJLUdjZEZyZWVCYXNpc0dGJEk/R3JlYXRlc3RGYWN0b3JpYWxGYWN0b3JpemF0aW9uR0YkSShIdXJ3aXR6R0YkSTFJc1NlbGZSZWNpcHJvY2FsR0YkSTJNaW5pbWFsUG9seW5vbWlhbEdGJEkwUERFVG9Qb2x5bm9taWFsR0YkSTBQb2x5bm9taWFsVG9QREVHRiRJMFNoaWZ0RXF1aXZhbGVudEdGJEk3U2hpZnRsZXNzRGVjb21wb3NpdGlvbkdGJEkoU2hvcnRlbkdGJEkoU2hvcnRlckdGJEklU29ydEdGJEkmU3BsaXRHRiRJJ1NwbGl0c0dGJEkqVHJhbnNsYXRlR0Yk
<Text-field style="Heading 3" layout="Heading 3">8.3.1. Polinomok.</Text-field> p1:=2*x^2+x+3; p2:=5*x^3+9; p1+p2; p1*p2; expand(%); LCgqJiIiIyIiIilJInhHNiJGJEYlRiVGJ0YlIiIkRiU= LCYqJiIiJiIiIilJInhHNiIiIiRGJUYlIiIqRiU= LCoqJiIiIyIiIilJInhHNiJGJEYlRiVGJ0YlIiM3RiUqJiIiJkYlKUYnIiIkRiVGJQ== KiYsKComIiIjIiIiKUkieEc2IkYlRiZGJkYoRiYiIiRGJkYmLCYqJiIiJkYmKUYoRipGJkYmIiIqRiZGJg== LC4qJiIjNSIiIilJInhHNiIiIiZGJUYlKiYiIz1GJSlGJyIiI0YlRiUqJkYpRiUpRiciIiVGJUYlKiYiIipGJUYnRiVGJSomIiM6RiUpRiciIiRGJUYlIiNGRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.2. Form\303\241lis hatv\303\241nysorok.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.3. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.4. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.5. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.6. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.7. Polinomf\303\274ggv\303\251nyek.</Font></Text-field> map(x->x mod 5,[0,1,2,3,4]); map(x->x^5 mod 5,[0,1,2,3,4]); NyciIiEiIiIiIiMiIiQiIiU= NyciIiEiIiIiIiMiIiQiIiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.8. A marad\303\251kos oszt\303\241s t\303\251tele polinomokra.</Font></Text-field> a:=5*x^4+9; b:=x^2+x+1; r:=rem(a,b,x); q:=quo(a,b,x); expand(q*b+r); LCYqJiIiJiIiIilJInhHNiIiIiVGJUYlIiIqRiU= LCgqJClJInhHNiIiIiMiIiJGKEYlRihGKEYo LCYiIioiIiIqJiIiJkYkSSJ4RzYiRiRGJA== LCYqJiIiJiIiIilJInhHNiIiIiNGJUYlKiZGJEYlRidGJSEiIg== LCYqJiIiJiIiIilJInhHNiIiIiVGJUYlIiIqRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.9. K\303\266vetkezm\303\251ny: gy\303\266kt\303\251nyez\305\221 lev\303\241laszt\303\241sa.</Font></Text-field> quo(x^3-1,x-1,x,'r'); r; LCgqJClJInhHNiIiIiMiIiJGKEYlRihGKEYo IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.10. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.11. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.12. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.13. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.14. Megjegyz\303\251s.</Font></Text-field> polydivisorx:=proc(a,b) local bb,db,r,dr,rr,rrr,q; if a=0 then return true fi; if b=0 then return false fi; bb:=lcoeff(b); db:=degree(b); r:=a; while degree(r)>=degree(b) do rr:=lcoeff(r); dr:=degree(r); rrr:=irem(rr,bb,'q'); if rrr<>0 then return false fi; r:=expand(r-q*b*x^(dr-db)); od; evalb(r=0); end; Zio2JEkiYUc2IkkiYkdGJTYpSSNiYkdGJUkjZGJHRiVJInJHRiVJI2RyR0YlSSNyckdGJUkkcnJyR0YlSSJxR0YlRiVGJUMpQCQvRiQiIiFPSSV0cnVlRyUqcHJvdGVjdGVkR0AkL0YmRjJPSSZmYWxzZUdGNT5GKC1JJ2xjb2VmZkdGNTYjRiY+RiktSSdkZWdyZWVHRjVGPT5GKkYkPyhGJSIiIkZDRiUxRj8tRkA2I0YqQyc+RiwtRjxGRj5GK0ZFPkYtLUklaXJlbUdGNTYlRixGKC5GLkAkMEYtRjJGOD5GKi1JJ2V4cGFuZEdGNTYjLCZGKkZDKihGLkZDRiZGQylJInhHRiUsJkYrRkNGKSEiIkZDRmVuLUkmZXZhbGJHRjU2Iy9GKkYyRiVGJUYl debug(polydivisorx); SS1wb2x5ZGl2aXNvcnhHNiI= polydivisorx(4*x^2-1,2*x+1); {--> enter polydivisorx, args = 4*x^2-1, 2*x+1 IiIj IiIi LCYqJiIiJSIiIilJInhHNiIiIiNGJUYlRiUhIiI= IiIl IiIj IiIh LCYiIiIhIiIqJiIiI0YjSSJ4RzYiRiNGJA== ISIj IiIi IiIh IiIh SSV0cnVlRyUqcHJvdGVjdGVkRw== <-- exit polydivisorx (now at top level) = true} SSV0cnVlRyUqcHJvdGVjdGVkRw== undebug(polydivisorx); SS1wb2x5ZGl2aXNvcnhHNiI= polydivisorx(4*x^2-1,2*x+1); SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.15. Megjegyz\303\251s: pszeudooszt\303\241</Font>s.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.16. Megjegyz\303\251s: Horner-elrendez\303\251s.</Font></Text-field> Horner:=proc(L::list,c) local i,LL,r; LL:=[]; if nops(L)=0 then return LL,0 fi; r:=L[nops(L)]; for i from nops(L)-1 to 1 by -1 do LL:=[r,op(LL)]; r:=L[i]+r*c; od; LL,r; end; Zio2JCdJIkxHNiJJJWxpc3RHJSpwcm90ZWN0ZWRHSSJjR0YmNiVJImlHRiZJI0xMR0YmSSJyR0YmRiZGJkMnPkYsNyJAJC8tSSVub3BzR0YoNiNGJSIiIU82JEYsRjY+Ri0mRiU2I0YzPyhGKywmRjMiIiJGPiEiIkY/Rj5JJXRydWVHRihDJD5GLDckRi0tSSNvcEdGKDYjRiw+Ri0sJiZGJTYjRitGPiomRi1GPkYpRj5GPjYkRixGLUYmRiZGJg== p:=x^3+4*x^2-3*x+7; CoefficientList(p,x); Horner(%,2); quo(p,x-2,x); rem(p,x-2,x); LCoqJClJInhHNiIiIiQiIiJGKComIiIlRigpRiUiIiNGKEYoKiZGJ0YoRiVGKCEiIiIiKEYo NyYiIighIiQiIiUiIiI= NiQ3JSIiKiIiJyIiIiIjRA== LCgqJClJInhHNiIiIiMiIiJGKComIiInRihGJUYoRigiIipGKA== IiNE convert(p,horner); LCYiIigiIiIqJiwmIiIkISIiKiYsJiIiJUYkSSJ4RzYiRiRGJEYsRiRGJEYkRixGJEYk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.17. Megjegyz\303\251s.</Font></Text-field> solve(x^2+1,x); NiReIyIiIiwkRiMhIiI= msolve(x^2+1,2); PCMvSSJ4RzYiIiIi msolve(x^2+1,3); msolve(x^2+1,5); NiQ8Iy9JInhHNiIiIiM8Iy9GJSIiJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.20. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.22. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.23. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.25. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.27. K<Font encoding="UTF-8">\303\266roszt\303\241</Font>si polinomok. </Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.28. Wilson t\303\251tele.</Font></Text-field> map(i->(i-1)!+1 mod i,[i$i=2..30]); Nz8iIiFGIyIiJEYjIiIiRiNGJUYlRiVGI0YlRiNGJUYlRiVGI0YlRiNGJUYlRiVGI0YlRiVGJUYlRiVGI0Yl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.29. Polinom algebrai deriv\303\241ltja.</Font></Text-field> p:=x^4+3*x^2+2*x+1; diff(p,x); LCoqJClJInhHNiIiIiUiIiJGKComIiIkRigpRiUiIiNGKEYoKiZGLEYoRiVGKEYoRihGKA== LCgqJiIiJSIiIilJInhHNiIiIiRGJUYlKiYiIidGJUYnRiVGJSIiI0Yl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.30. Megjegyz\303\251s.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.31. T\303\251tel.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.32. K\303\266vetkezm\303\251ny.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.33. T\303\266bbsz\303\266r\303\266s gy\303\266k\303\266k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.34. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.35. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.36. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.37. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.38. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.39. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.40. Irreducibilis polinomok \303\251s testb\305\221v\303\255t\303\251sek.</Font></Text-field> Split(p,x); KiosKkkieEc2IiIiIi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCoqJClJI19aR0YpIiIlRiZGJiomIiIkRiYpRjAiIiNGJkYmKiZGNUYmRjBGJkYmRiZGJkYmLUYoNiMsLiokKUYwRjNGJkYmKiZGJ0YmRjRGJkYmKiYsJkYzRiYqJClGJ0Y1RiZGJkYmRjBGJkYmRjVGJiomRjNGJkYnRiZGJiokKUYnRjNGJkYmRiYtRig2IywuKiRGNEYmRiYqJiwmRidGJkY3RiZGJkYwRiZGJkYzRiZGP0YmKiZGN0YmRidGJkYmKiQpRjdGNUYmRiZGJkYmLCZGJEYmRjchIiJGJiwmRiRGJkYnRk5GJiwmRiRGJkZERk5GJg== p:=x^4; Nextpoly(p,x) mod 2; Nextpoly(%,x) mod 2; Prevpoly(%,x) mod 2; KiQpSSJ4RzYiIiIlIiIi LCYqJClJInhHNiIiIiUiIiJGKEYoRig= LCYqJClJInhHNiIiIiUiIiJGKEYlRig= LCYqJClJInhHNiIiIiUiIiJGKEYoRig= p:=x^4; Nextprime(p,x) mod 2; Nextprime(%,x) mod 2; Prevprime(%,x) mod 2; KiQpSSJ4RzYiIiIlIiIi LCgqJClJInhHNiIiIiUiIiJGKEYlRihGKEYo LCgqJClJInhHNiIiIiUiIiJGKCokKUYlIiIkRihGKEYoRig= LCgqJClJInhHNiIiIiUiIiJGKEYlRihGKEYo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.41. P\303\251ld\303\241k.</Font></Text-field> Split(x^2+1,x); KiYsJkkieEc2IiIiIi1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYqJClJI19aR0YpIiIjRiZGJkYmRiYhIiJGJiwmRiRGJkYnRiZGJg== p:=x^2+x+1; modpol(x^8+4*x^2,p,x,2); modpol(1/%,p,x,2); modpol(%*%%,p,x,2); LCgqJClJInhHNiIiIiMiIiJGKEYlRihGKEYo LCZJInhHNiIiIiJGJUYl SSJ4RzYi IiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.42. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.43. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.44. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.45. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.46. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.47. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.48. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.49. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.50. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.51. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.52. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.53. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.54. V\303\251ges testek elemsz\303\241</Font>ma.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.55. Megjegyz\303\251sek.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.56. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.57. Alkalmaz<Font encoding="UTF-8">\303\241</Font>s: a Rijndael <Font encoding="UTF-8">\303\251s AES blokkrejtjelz\305\221</Font>k.</Text-field> lgn:=8; n:=2^lgn-1; RijndaelPoly:=Nextprime(Z^lgn,Z) mod 2; alpha:=Z; IiIp IiRiIw== LCwqJClJIlpHNiIiIikiIiJGKCokKUYlIiIlRihGKCokKUYlIiIkRihGKEYlRihGKEYo SSJaRzYi C:=Matrix([[1,0,0,0,1,1,1,1],[1,1,0,0,0,1,1,1],[1,1,1,0,0,0,1,1],[1,1,1,1,0,0,0,1],[1,1,1,1,1,0,0,0],[0,1,1,1,1,1,0,0],[0,0,1,1,1,1,1,0],[0,0,0,1,1,1,1,1]]);c:=Vector([1,1,0,0,0,1,1,0]); LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKltzIjQ8 LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqU0A/cSI= with(LinearAlgebra); 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 Cinv:=MatrixInverse(C) mod 2; LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKiEpPkhwIg== S:=proc(x) local i,xx; global RijndaelPoly,C,c; xx:=convert(x,base,2); xx:=add(xx[i]*Z^(i-1),i=1..nops(xx)); if xx<>0 then xx:=modpol(1/xx,RijndaelPoly,Z,2) else xx:=modpol(xx,RijndaelPoly,Z,2) fi; xx:=CoefficientList(xx,Z); while nops(xx)<8 do xx:=[op(xx),0] od; xx:=convert(xx,Vector); xx:=Multiply(C,xx) mod 2; xx:=Add(xx,c) mod 2; add(xx[i]*2^(i-1),i=1..8); end; S(0); S(1); S(2); 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 IiMqKg== IiRDIg== IiQ+Ig== Sinv:=proc(x) local i,xx; global RijndaelPoly,Cinv,c; xx:=convert(x,base,2); while nops(xx)<8 do xx:=[op(xx),0] od; xx:=convert(xx,Vector); xx:=Add(xx,c,1,-1) mod 2; xx:=Multiply(Cinv,xx) mod 2; xx:=add(xx[i]*Z^(i-1),i=1..8); if xx<>0 then xx:=modpol(1/xx,RijndaelPoly,Z,2) fi; xx:=CoefficientList(xx,Z); add(xx[i]*2^(i-1),i=1..nops(xx)); end; Sinv(99); Sinv(124); Sinv(119); 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 IiIh IiIi IiIj X:=proc(x,y,b) local u,v,i,xx,yy; # bitwise xor in b bit length xx:=convert(x,base,2); yy:=convert(y,base,2); while nops(xx)<b do xx:=[op(xx),0] od; while nops(yy)<b do yy:=[op(yy),0] od; xx:=zip((u,v)->u+v mod 2,xx,yy); add(xx[i]*2^(i-1),i=1..b); end; X(5,3,4); Zio2JUkieEc2IkkieUdGJUkiYkdGJTYnSSJ1R0YlSSJ2R0YlSSJpR0YlSSN4eEdGJUkjeXlHRiVGJUYlQyg+RiwtSShjb252ZXJ0RyUqcHJvdGVjdGVkRzYlRiRJJWJhc2VHRiUiIiM+Ri0tRjE2JUYmRjRGNT8oRiUiIiJGOkYlMi1JJW5vcHNHRjI2I0YsRic+Riw3JC1JI29wR0YyRj4iIiE/KEYlRjpGOkYlMi1GPTYjRi1GJz5GLTckLUZCRkdGQz5GLC1JJHppcEdJKF9zeXNsaWJHRiU2JWYqNiRGKUYqRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSRtb2RHRiU2JCwmRilGOkYqRjpGNUYlRiVGJUYsRi0tSSRhZGRHRjI2JComJkYsNiNGK0Y6KUY1LCZGK0Y6RjohIiJGOi9GKztGOkYnRiVGJUYl IiIn Word2Bytes:=proc(x) local xx; xx:=convert(x,base,256); while nops(xx)<4 do xx:=[op(xx),0] od; [xx[4],xx[3],xx[2],xx[1]]; end; Bytes2Word:=proc(x) local i; add(x[i]*256^(4-i),i=1..4) end; Bytes2Word([0,1,2,3]); Word2Bytes(%); Zio2I0kieEc2IjYjSSN4eEdGJUYlRiVDJT5GJy1JKGNvbnZlcnRHJSpwcm90ZWN0ZWRHNiVGJEklYmFzZUdGJSIkYyM/KEYlIiIiRjFGJTItSSVub3BzR0YsRiYiIiU+Ric3JC1JI29wR0YsRiYiIiE3JiZGJzYjRjUmRic2IyIiJCZGJzYjIiIjJkYnNiNGMUYlRiVGJQ== Zio2I0kieEc2IjYjSSJpR0YlRiVGJS1JJGFkZEclKnByb3RlY3RlZEc2JComJkYkRiYiIiIpIiRjIywmIiIlRi5GJyEiIkYuL0YnO0YuRjJGJUYlRiU= IiZeZyc= NyYiIiEiIiIiIiMiIiQ= K:=["00010203","05060708","0A0B0C0D","0F101112"];K:=map(x->convert(x,decimal,hex),K); NyZRKTAwMDEwMjAzNiJRKTA1MDYwNzA4RiRRKTBBMEIwQzBERiRRKTBGMTAxMTEyRiQ= NyYiJl5nJyIpJzQiRyUpIipUaFxvIiIqJz02RkQ= RijndaelKeys:=proc(K,kk) local i,t,x,tt,k,KK,R; global RijndaelPoly; k:=nops(K); KK:=K; R:=1; for i from k to kk-1 do t:=KK[i]; if k>6 and (i mod k=4) then t:=Word2Bytes(t); t:=map(x->S(x),t); t:=Bytes2Word(t); fi; if i mod k=0 then t:=Word2Bytes(t); t:=[t[2],t[3],t[4],t[1]]; t:=map(x->S(x),t); tt:=CoefficientList(R,Z); tt:=add(tt[i]*2^(i-1),i=1..nops(tt)); tt:=X(t[1],tt,8); t:=[tt,t[2],t[3],t[4]]; R:=modpol(R*Z,RijndaelPoly,Z,2); t:=Bytes2Word(t); fi; t:=X(t,KK[i-k+1],32); KK:=[op(KK),t]; od; KK; end; RijndaelKeys(K,8); 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 NyoiJl5nJyIpJzQiRyUpIipUaFxvIiIqJz02RkQiK1xAVDlNIitMXihbWSQiKzcoKm8oSCQiK1FKPTtN SubBytes:=proc(L) local x,y; global S; map(x->map(y->S(y),x),L); end; Zio2I0kiTEc2IjYkSSJ4R0YlSSJ5R0YlRiVGJS1JJG1hcEclKnByb3RlY3RlZEc2JGYqNiNGJ0YlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUYqNiRmKjYjRihGJUYvRiUtSSJTR0YlRjVGJUYlRiVGJ0YlRiVGJUYkRiU2I0Y3RiU= SubBytesinv:=proc(L) local x,y; global Sinv; map(x->map(y->Sinv(y),x),L); end; Zio2I0kiTEc2IjYkSSJ4R0YlSSJ5R0YlRiVGJS1JJG1hcEclKnByb3RlY3RlZEc2JGYqNiNGJ0YlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUYqNiRmKjYjRihGJUYvRiUtSSVTaW52R0YlRjVGJUYlRiVGJ0YlRiVGJUYkRiU2I0Y3RiU= ShiftRow:=proc(L) local b,i,d,LL,LLL; b:=nops(L); LLL:=[]; if b>6 then d:=1 else d:=0 fi; for i to b do LL:=[L[i][1],L[1+(i mod b)][2],L[1+(i+1+d mod b)][3],L[1+(i+2+d mod b)][4]]; LLL:=[op(LLL),LL]; od; LLL; end; ShiftRow([[0,1,2,3],[10,11,12,13],[20,21,22,23],[30,31,32,33]]); Zio2I0kiTEc2IjYnSSJiR0YlSSJpR0YlSSJkR0YlSSNMTEdGJUkkTExMR0YlRiVGJUMnPkYnLUklbm9wc0clKnByb3RlY3RlZEdGIz5GKzciQCUyIiInRic+RikiIiI+RikiIiE/KEYoRjdGN0YnSSV0cnVlR0YwQyQ+Rio3JiYmRiQ2I0YoNiNGNyYmRiQ2IywmRjdGNy1JJG1vZEdGJTYkRihGJ0Y3NiMiIiMmJkYkNiMsJkY3RjctRkg2JCwoRihGN0Y3RjdGKUY3RidGNzYjIiIkJiZGJDYjLCZGN0Y3LUZINiQsKEYoRjdGS0Y3RilGN0YnRjc2IyIiJT5GKzckLUkjb3BHRjA2I0YrRipGK0YlRiVGJQ== NyY3JiIiISIjNiIjQSIjTDcmIiM1IiNAIiNLIiIkNyYiIz8iI0oiIiMiIzg3JiIjSSIiIiIjNyIjQg== ShiftRowinv:=proc(L) local b,i,d,LL,LLL; b:=nops(L); LLL:=[]; if b>6 then d:=1 else d:=0 fi; for i to b do LL:=[L[i][1],L[1+(i-2 mod b)][2],L[1+(i-3-d mod b)][3],L[1+(i-4-d mod b)][4]]; LLL:=[op(LLL),LL]; od; LLL; end; ShiftRowinv([[0,1,2,3],[10,11,12,13],[20,21,22,23],[30,31,32,33]]); Zio2I0kiTEc2IjYnSSJiR0YlSSJpR0YlSSJkR0YlSSNMTEdGJUkkTExMR0YlRiVGJUMnPkYnLUklbm9wc0clKnByb3RlY3RlZEdGIz5GKzciQCUyIiInRic+RikiIiI+RikiIiE/KEYoRjdGN0YnSSV0cnVlR0YwQyQ+Rio3JiYmRiQ2I0YoNiNGNyYmRiQ2IywmRjdGNy1JJG1vZEdGJTYkLCZGKEY3IiIjISIiRidGNzYjRksmJkYkNiMsJkY3RjctRkg2JCwoRihGNyIiJEZMRilGTEYnRjc2I0ZVJiZGJDYjLCZGN0Y3LUZINiQsKEYoRjciIiVGTEYpRkxGJ0Y3NiNGaG4+Ris3JC1JI29wR0YwNiNGK0YqRitGJUYlRiU= NyY3JiIiISIjSiIjQSIjODcmIiM1IiIiIiNLIiNCNyYiIz8iIzYiIiMiI0w3JiIjSSIjQCIjNyIiJA== normalizepolyzZ:=proc(p) local i; global RijndaelPoly; CoefficientList(expand(p) mod 2,z); map(x->modpol(x,RijndaelPoly,Z,2),%); add(%[i]*z^(i-1),i=1..nops(%)); sort(%); end; Zio2I0kicEc2IjYjSSJpR0YlRiVGJUMmLV9JMFBvbHlub21pYWxUb29sc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUkwQ29lZmZpY2llbnRMaXN0R0YlNiQtSSRtb2RHRiU2JC1JJ2V4cGFuZEdGLUYjIiIjSSJ6R0YlLUkkbWFwR0YtNiRmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSdtb2Rwb2xHRiw2JkY9SS1SaWpuZGFlbFBvbHlHRiVJIlpHRiVGNkYlRiVGJUkiJUdGJS1JJGFkZEdGLTYkKiYmRkZGJiIiIilGNywmRidGTEZMISIiRkwvRic7RkwtSSVub3BzR0YtNiNGRi1JJXNvcnRHRi1GVEYlNiNGREYl RijndaelMixPoly:=(Z+1)*z^3+z^2+z+Z; MixMul:=proc(L) local p,LL,x,i; global RijndaelPoly,RijndaelMixPoly; LL:=map(x->convert(x,base,2),L); LL:=map(x->add(x[i]*Z^(i-1),i=1..nops(x)),LL); p:=add(LL[i]*z^(i-1),i=1..4); p:=p*RijndaelMixPoly; p:=normalizepolyzZ(p); while degree(p,z)>=4 do p:=normalizepolyzZ(p-lcoeff(p,z)*z^(degree(p,z)-4)*(z^4+1) mod 2); od; LL:=CoefficientList(p,z); while nops(LL)<4 do LL:=[op(LL),0] od; map(x->subs(Z=2,x),LL); end; MixMul([1,1,1,1]); MixMul([219,19,83,69]); MixMul([242,10,34,92]); MixMul([198,198,198,198]); MixMul([212,212,212,213]); MixMul([45,38,49,76]); LCoqJiwmSSJaRzYiIiIiRidGJ0YnKUkiekdGJiIiJEYnRicqJClGKSIiI0YnRidGKUYnRiVGJw== 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 NyYiIiJGI0YjRiM= NyYiJFUiIiN4IiRoIiIkKT0= NyYiJGYiIiQ/IyIjKSkiJGQi NyYiJCk+RiNGI0Yj NyYiJDgjRiMiJDojIiQ5Iw== NyYiI3giJEUiIiQqPSIkWyM= RijndaelMixPolyinv:=(Z^3+Z+1)*z^3+(Z^3+Z^2+1)*z^2+(Z^3+1)*z+Z^3+Z^2+Z; MixMulinv:=proc(L) local p,LL,x,i; global RijndaelPoly,RijndaelMixPolyinv; LL:=map(x->convert(x,base,2),L); LL:=map(x->add(x[i]*Z^(i-1),i=1..nops(x)),LL); p:=add(LL[i]*z^(i-1),i=1..4); p:=p*RijndaelMixPolyinv; p:=normalizepolyzZ(p); while degree(p,z)>=4 do p:=normalizepolyzZ(p-lcoeff(p,z)*z^(degree(p,z)-4)*(z^4+1) mod 2); od; LL:=CoefficientList(p,z); while nops(LL)<4 do LL:=[op(LL),0] od; map(x->subs(Z=2,x),LL); end; MixMulinv([1,1,1,1]); MixMulinv([142,77,161,188]); MixMulinv([159,220,88,157]); MixMulinv([198,198,198,198]); MixMulinv([213,213,215,214]); MixMulinv([77,126,189,248]); LC4qJiwoKiQpSSJaRzYiIiIkIiIiRipGJ0YqRipGKkYqKUkiekdGKEYpRipGKiomLChGJUYqKiQpRiciIiNGKkYqRipGKkYqKUYsRjFGKkYqKiYsJkYlRipGKkYqRipGLEYqRipGJUYqRi9GKkYnRio= 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 NyYiIiJGI0YjRiM= NyYiJD4jIiM+IiMkKSIjcA== NyYiJFUjIiM1IiNNIiMjKg== NyYiJCk+RiNGI0Yj NyYiJDcjRiNGIyIkOCM= NyYiI1giI1EiI1wiI3c= Rijndael:=proc(M::list(posint),K::list(posint),r::posint) local i,j,k,m,KK,MM,MMM; k:=nops(K); m:=nops(M); MMM:=[]; KK:=RijndaelKeys(K,m*(r+1)); for j to m do MMM:=[op(MMM),X(KK[j],M[j],32)]; od; for i from 2 to r do MM:=map(x->Word2Bytes(x),MMM); MM:=SubBytes(MM); MM:=ShiftRow(MM); MM:=map(x->MixMul(x),MM); MM:=map(x->Bytes2Word(x),MM); MMM:=[]; for j to m do MMM:=[op(MMM),X(KK[(i-1)*m+j],MM[j],32)]; od; od; MM:=map(x->Word2Bytes(x),MMM); MM:=SubBytes(MM); MM:=ShiftRow(MM); MM:=map(x->Bytes2Word(x),MM); MMM:=[]; for j to m do MMM:=[op(MMM),X(KK[r*m+j],MM[j],32)]; od; MMM; end; M:=["506812A4","5F08C889","B97F5980","038B8359"]; M:=map(x->convert(x,decimal,hex),M); Rijndael(M,K,10); 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 NyZRKTUwNjgxMkE0NiJRKTVGMDhDODg5RiRRKUI5N0Y1OTgwRiRRKTAzOEI4MzU5RiQ= NyYiKyd6KCoqWzgiK1g2VCVmIiIrVzQ4N0oiKXhaWmY= NyYiK2Z5JSpSTyIrQHkyIT4jIio3cV83IiIrdCNmMUQl Rijndaelinv:=proc(M::list(posint),K::list(posint),r::posint) local i,j,k,m,KK,MM,MMM; k:=nops(K); m:=nops(M); MMM:=[]; KK:=RijndaelKeys(K,m*(r+1)); for j to m do MMM:=[op(MMM),X(KK[r*m+j],M[j],32)]; od; MM:=map(x->Word2Bytes(x),MMM); MM:=ShiftRowinv(MM); MM:=SubBytesinv(MM); MM:=map(x->Bytes2Word(x),MM); for i from r to 2 by -1 do MMM:=[]; for j to m do MMM:=[op(MMM),X(KK[(i-1)*m+j],MM[j],32)]; od; MM:=map(x->Word2Bytes(x),MMM); MM:=map(x->MixMulinv(x),MM); MM:=ShiftRowinv(MM); MM:=SubBytesinv(MM); MM:=map(x->Bytes2Word(x),MM); od; MMM:=[]; for j to m do MMM:=[op(MMM),X(KK[j],MM[j],32)]; od; MMM; end; CM:=["D8F53253","8289EF7D","06B506A4","FD5BE9C9"]; CM:=map(x->convert(x,decimal,hex),CM); Rijndaelinv(CM,K,10); M:=["506812A4","5F08C889","B97F5980","038B8359"]; M:=map(x->convert(x,decimal,hex),M); 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 NyZRKUQ4RjUzMjUzNiJRKTgyODlFRjdERiRRKTA2QjUwNkE0RiRRKUZENUJFOUM5RiQ= NyYiK2Z5JSpSTyIrQHkyIT4jIio3cV83IiIrdCNmMUQl NyYiKyd6KCoqWzgiK1g2VCVmIiIrVzQ4N0oiKXhaWmY= NyZRKTUwNjgxMkE0NiJRKTVGMDhDODg5RiRRKUI5N0Y1OTgwRiRRKTAzOEI4MzU5RiQ= NyYiKyd6KCoqWzgiK1g2VCVmIiIrVzQ4N0oiKXhaWmY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.58. T\303\251tel.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.59. T\303\251tel.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.60. K\303\266vetkezm\303\251ny.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.61. K\303\266vetkezm\303\251ny.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.62. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.63. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.64. Irreducibilis polinomok.</Text-field> factor(x^3-1); factor(6*x^2+12*x+12); KiYsJkkieEc2IiIiIkYmISIiRiYsKCokKUYkIiIjRiZGJkYkRiZGJkYmRiY= LCgqJiIiJyIiIilJInhHNiIiIiNGJUYlKiYiIzdGJUYnRiVGJUYrRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.65. Primit\303\255v polinomok.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.66. Gauss lemm\303\241ja.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*8.3.67. Lemma.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.68. Gauss t\303\251tele.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.69. Legnagyobb k\303\266z\303\266s oszt\303\263</Font> sz<Font encoding="UTF-8">\303\241</Font>mol<Font encoding="UTF-8">\303\241</Font>sa Gauss-g<Font encoding="UTF-8">y\305\261r\305\261</Font> feletti polinomg<Font encoding="UTF-8">y\305\261r\305\261 eset\303\251n.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.70. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.71. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.72. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.73. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.74. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.75. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.76. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.77. Sch\303\266nemann-Eisenstein-t\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.78. K\303\266vetkezm\303\251ny.</Font></Text-field> factor(x^4+7); LCYqJClJInhHNiIiIiUiIiJGKCIiKEYo (x^5-1)/(x-1); simplify(%); Translate(%,x,1); KiYsJiokKUkieEc2IiIiJiIiIkYpRikhIiJGKSwmRiZGKUYpRipGKg== LCwqJClJInhHNiIiIiUiIiJGKCokKUYlIiIkRihGKCokKUYlIiIjRihGKEYlRihGKEYo LCwiIiYiIiIqJiIjNUYkSSJ4RzYiRiRGJComRiZGJClGJyIiI0YkRiQqJkYjRiQpRiciIiRGJEYkKiQpRiciIiVGJEYk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.79. Megjegyz\303\251sek.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.80. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.81. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.82. Lagrange-interpol\303\241ci\303\263.</Font></Text-field> with(CurveFitting); NypJKEJTcGxpbmVHNiJJLUJTcGxpbmVDdXJ2ZUdGJEksSW50ZXJhY3RpdmVHRiRJLUxlYXN0U3F1YXJlc0dGJEk4UG9seW5vbWlhbEludGVycG9sYXRpb25HRiRJNlJhdGlvbmFsSW50ZXJwb2xhdGlvbkdGJEknU3BsaW5lR0YkSTRUaGllbGVJbnRlcnBvbGF0aW9uR0Yk PolynomialInterpolation([[0,2],[1,4],[3,7],[4,5]],x); LCoqJiMiIiIiIiVGJSlJInhHNiIiIiRGJSEiIiomIyIiJiIiJ0YlKUYoIiIjRiVGJSomIyIjPCIjN0YlRihGJUYlRjFGJQ== PolynomialInterpolation([[0,2],[1,4],[3,7],[4,5]],x,form=Lagrange); LCoqKiMiIiIiIidGJSwmSSJ4RzYiRiVGJSEiIkYlLCZGKEYlIiIkRipGJSwmRihGJSIiJUYqRiVGKioqIyIiI0YsRiVGKEYlRitGJUYtRiVGJSoqIyIiKEYmRiVGKEYlRidGJUYtRiVGKioqIyIiJiIjN0YlRihGJUYnRiVGK0YlRiU=
<Text-field style="Heading 3" layout="Heading 3">8.3.83. Titokmegoszt<Font encoding="UTF-8">\303\241s.</Font></Text-field> p:=nextprime(10^50); t:=12345678901234567890123456789012345678901234567890; P:=t+94536789321234890676867542309176345623498734562349*x+79888998766657439865923165346578934524359870127935*x^2 mod p; IlReLCsrKysrKysrKysrKysrKysrKysrKysrIg== IlMhKnljTTchKnljTTchKnljTTchKnljTTchKnljTTc= LCgiUyEqeWNNNyEqeWNNNyEqeWNNNyEqeWNNNyEqeWNNNyIiIiomIlNcQmNNKClcQmNNdyI0QmFub24hKltCQCQqeU9YKkYkSSJ4RzYiRiRGJComIlNOejdxKWZWX00qeWxNbEojZicpUnVsbSgpKiopKSl6RiQpRiciIiNGJEYk y1:=subs(x=1,P) mod p; y2:=subs(x=2,P) mod p; y3:=subs(x=3,P) mod p; y4:=subs(x=4,P) mod p; y5:=subs(x=5,P) mod p; IlNCIWUjUilmbiNlaW5aV2tUIkhWKSpvNyopcFlybik= IlN0Tj8lPVFMLXYybyR6LTdiMigzVEw1RUR2NCM= IlNVW1NwaWpvS3peZCQ9ZE1Tcik+Y1t3TnEmXCI= IlNJPSdbNGFFYyFvISk0ZHJVT18lb0pwX2sib3Jv IlMnUXUwbSJSMHBWbiQqKj5JU0QjeixYUW4lZmFBKQ== Interp([1,3,5],[y1,y3,y5],x) mod p; LCgiUyEqeWNNNyEqeWNNNyEqeWNNNyEqeWNNNyEqeWNNNyIiIiomIlNcQmNNKClcQmNNdyI0QmFub24hKltCQCQqeU9YKkYkSSJ4RzYiRiRGJComIlNOejdxKWZWX00qeWxNbEojZicpUnVsbSgpKiopKSl6RiQpRiciIiNGJEYk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.84. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.85. Kronecker-elj\303\241r\303\241s.</Font></Text-field> with(combinat); 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 with(numtheory); 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 Kroneckerfactorx:=proc(p) local d,D,pp,i,X,Y,v,V,c,q,y; pp:=expand(p); if pp=0 then return [0] fi; if pp=1 then return [] fi; if pp=-1 then return [-1] fi; D:=degree(pp); X:=[]; Y:=[]; i:=0; while nops(X)<=floor(D/2) do v:=subs(x=i,pp); if v=0 then return [x-i,op(Kroneckerfactorx(quo(pp,x-i,x)))] fi; V:=divisors(v); V:=V union map(y->-y,V); V:=[op(V)]; V:=sort(V,(u,v)->abs(u)<abs(v)); X:=[op(X),i]; Y:=[op(Y),V]; if i<=0 then i:=1-i else i:=-i fi; od; for d from 0 while D>=2*d do y:=cartprod(Y[1..d+1]); while not y[finished] do q:=PolynomialInterpolation(X[1..d+1],y[nextvalue](),x); if q=1 or q=-1 then next fi; if polydivisorx(pp,q) then return [q,op(Kroneckerfactorx(quo(pp,q,x)))] fi; od; od; [pp]; end; 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 debug(Kroneckerfactorx); STFLcm9uZWNrZXJmYWN0b3J4RzYi p:=x^5-2*x^4-2*x^3+4*x^2+x-2; Kroneckerfactorx(p); LC4qJClJInhHNiIiIiYiIiJGKComIiIjRigpRiUiIiVGKCEiIiomRipGKClGJSIiJEYoRi0qJkYsRigpRiVGKkYoRihGJUYoRipGLQ== {--> enter Kroneckerfactorx, args = x^5-2*x^4-2*x^3+4*x^2+x-2 LC4qJClJInhHNiIiIiYiIiJGKComIiIjRigpRiUiIiVGKCEiIiomRipGKClGJSIiJEYoRi0qJkYsRigpRiVGKkYoRihGJUYoRipGLQ== IiIm NyI= NyI= IiIh ISIj PCQiIiIiIiM= PCYhIiMhIiIiIiIiIiM= NyYhIiMhIiIiIiIiIiM= NyYiIiIhIiIiIiMhIiM= NyMiIiE= NyM3JiIiIiEiIiIiIyEiIw== IiIi IiIh {--> enter Kroneckerfactorx, args = x^4-x^3-3*x^2+x+2 LCwqJClJInhHNiIiIiUiIiJGKCokKUYlIiIkRighIiIqJkYrRigpRiUiIiNGKEYsRiVGKEYvRig= IiIl NyI= NyI= IiIh IiIj PCQiIiIiIiM= PCYhIiMhIiIiIiIiIiM= NyYhIiMhIiIiIiIiIiM= NyYiIiIhIiIiIiMhIiM= NyMiIiE= NyM3JiIiIiEiIiIiIyEiIw== IiIi IiIh {--> enter Kroneckerfactorx, args = x^3-3*x-2 LCgqJClJInhHNiIiIiQiIiJGKComRidGKEYlRighIiIiIiNGKg== IiIk NyI= NyI= IiIh ISIj PCQiIiIiIiM= PCYhIiMhIiIiIiIiIiM= NyYhIiMhIiIiIiIiIiM= NyYiIiIhIiIiIiMhIiM= NyMiIiE= NyM3JiIiIiEiIiIiIyEiIw== IiIi ISIl PCUiIiIiIiMiIiU= PCghIiUhIiMhIiIiIiIiIiMiIiU= NyghIiUhIiMhIiIiIiIiIiMiIiU= NygiIiIhIiIiIiMhIiMiIiUhIiU= NyQiIiEiIiI= NyQ3JiIiIiEiIiIiIyEiIzcoRiRGJUYmRiciIiUhIiU= ISIi 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 IiIi ISIi IiIj ISIj 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 IiIi LCYqJiIiIyIiIkkieEc2IkYlISIiRiVGJQ== LCZJInhHNiIiIiJGJUYl {--> enter Kroneckerfactorx, args = x^2-x-2 LCgqJClJInhHNiIiIiMiIiJGKEYlISIiRidGKQ== IiIj NyI= NyI= IiIh ISIj PCQiIiIiIiM= PCYhIiMhIiIiIiIiIiM= NyYhIiMhIiIiIiIiIiM= NyYiIiIhIiIiIiMhIiM= NyMiIiE= NyM3JiIiIiEiIiIiIyEiIw== IiIi ISIj PCQiIiIiIiM= PCYhIiMhIiIiIiIiIiM= NyYhIiMhIiIiIiIiIiM= NyYiIiIhIiIiIiMhIiM= NyQiIiEiIiI= NyQ3JiIiIiEiIiIiIyEiI0Yj ISIi 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 IiIi ISIi IiIj ISIj 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 IiIi LCYqJiIiIyIiIkkieEc2IkYlISIiRiVGJQ== LCZJInhHNiIiIiJGJUYl {--> enter Kroneckerfactorx, args = x-2 LCZJInhHNiIiIiIiIiMhIiI= IiIi NyI= NyI= IiIh ISIj PCQiIiIiIiM= PCYhIiMhIiIiIiIiIiM= NyYhIiMhIiIiIiIiIiM= NyYiIiIhIiIiIiMhIiM= NyMiIiE= NyM3JiIiIiEiIiIiIyEiIw== IiIi 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 IiIi ISIi IiIj ISIj NyMsJkkieEc2IiIiIiIiIyEiIg== <-- exit Kroneckerfactorx (now in Kroneckerfactorx) = [x-2]} <-- exit Kroneckerfactorx (now in Kroneckerfactorx) = [x+1, x-2]} <-- exit Kroneckerfactorx (now in Kroneckerfactorx) = [x+1, x+1, x-2]} <-- exit Kroneckerfactorx (now in Kroneckerfactorx) = [x-1, x+1, x+1, x-2]} <-- exit Kroneckerfactorx (now at top level) = [x-1, x-1, x+1, x+1, x-2]} NycsJkkieEc2IiIiIkYmISIiRiMsJkYkRiZGJkYmRigsJkYkRiYiIiNGJw== undebug(Kroneckerfactorx); STFLcm9uZWNrZXJmYWN0b3J4RzYi Kroneckerfactorx(6*x^2+12*x+12); NyUiIiMiIiQsKCokKUkieEc2IkYjIiIiRioqJkYjRipGKEYqRipGI0Yq LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.86. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.87. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.88. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->8.3.89. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.90. Cardano-k\303\251plet.</Font></Text-field> p:='p'; q:='q'; solve(x^3+p*x+q=0,x); SSJwRzYi SSJxRzYi NiUsJiomIyIiIiIiJ0YmKSwmKiYiJDMiRiZJInFHNiJGJiEiIiomIiM3RiYpLCYqJkYwRiYpSSJwR0YtIiIkRiZGJiomIiMiKUYmKUYsIiIjRiZGJiNGJkY6RiZGJiNGJkY2RiZGJiooRjpGJkY1RiZGKEYuRi4sKComI0YmRjBGJkYoRiZGLiomRjVGJkYoRi5GJiooKiZGO0YmXiNGJkYmRiYpRjZGO0YmLCZGJEYmRj1GJkYmRiYsKEY/Ri5GQUYmKigsJEZDRiZGJkZFRiZGRkYmRi4= r:='r'; solve(x^4+p*x^2+q*x+r=0,x); SSJyRzYi LUknUm9vdE9mRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsKiokKUkjX1pHRiQiIiUiIiJGLiomSSJwR0YnRi4pRiwiIiNGLkYuKiZJInFHRidGLkYsRi5GLkkickdGJ0Yu LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*8.3.91. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.92. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.93. Testb\305\221v\303\255t\303\251sek.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.94. T\303\251tel.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.95. Feladat: t<Font encoding="UTF-8">estb\305\221v\303\255t\303\251sek.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.96. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.97. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.98. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.99. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.100. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.101. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.102. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.103. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.104. Feladat: szerkeszthet<Font encoding="UTF-8">\305\221s\303\251g.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.105. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.106. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.107. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.108. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.109. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.110. V\303\251ges testek alapt\303\251tele.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.111. Wedderburn t\303\251tele.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.112. Polinomfaktoriz<Font encoding="UTF-8">\303\241</Font>l<Font encoding="UTF-8">\303\241s v\303\251ges testek felett.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.113. Magasabb fok<Font encoding="UTF-8">\303\272</Font> kongruenci<Font encoding="UTF-8">\303\241</Font>k.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.114. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.115. Hensel-lemma.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.116. Megjegyz\303\251s.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.117. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.118. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.119. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.120. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.121. Racion\303\241lis t\303\266rtf\303\274ggv\303\251nyek.</Font></Text-field> r1:=(z^3-1)/(z^2-1); r1:=simplify(r1); r2:=simplify((z^4-1)/(z^3-1)); r1*r2; KiYsJiokKUkiekc2IiIiJCIiIkYpRikhIiJGKSwmKiQpRiYiIiNGKUYpRilGKkYq KiYsKCokKUkiekc2IiIiIyIiIkYpRiZGKUYpRilGKSwmRiZGKUYpRikhIiI= KiYsKiokKUkiekc2IiIiJCIiIkYpKiQpRiYiIiNGKUYpRiZGKUYpRilGKSwoRipGKUYmRilGKUYpISIi KiYsJkkiekc2IiIiIkYmRiYhIiIsKiokKUYkIiIkRiZGJiokKUYkIiIjRiZGJkYkRiZGJkYmRiY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.122. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.123. Parci\303\241lis t\303\266rtekre bont\303\241s.</Font></Text-field> f:=1/(x^4-x^2); convert(f,parfrac,x); KiQsJiokKUkieEc2IiIiJSIiIkYpKiQpRiYiIiNGKSEiIkYt LCgqJClJInhHNiIiIiMhIiJGKComIyIiIkYnRissJkYlRitGK0YoRihGKyomRipGKywmRiVGK0YrRitGKEYo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.124. K\303\266vetkezm\303\251ny.</Font></Text-field> f:=36/(x^5-2*x^4-2*x^3+4*x^2+x-2); convert(f,parfrac,x); LCQqJiIjTyIiIiwuKiQpSSJ4RzYiIiImRiVGJSomIiIjRiUpRikiIiVGJSEiIiomRi1GJSlGKSIiJEYlRjAqJkYvRiUpRilGLUYlRiVGKUYlRi1GMEYwRiU= LCoqJiIiJCIiIiksJkkieEc2IkYlRiVGJSIiIyEiIkYrKiYiIiVGJUYnRitGKyomIiIqRiUpLCZGKEYlRiVGK0YqRitGKyomRi1GJSwmRihGJUYqRitGK0Yl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.125. K\303\266vetkezm\303\251ny.</Font></Text-field> f:=(x^5+1)/(x^4-x^2); convert(f,parfrac,x); KiYsJiokKUkieEc2IiIiJiIiIkYpRilGKUYpLCYqJClGJiIiJUYpRikqJClGJiIiI0YpISIiRjE= LChJInhHNiIiIiIqJClGIyIiIyEiIkYpKiQsJkYjRiVGJUYpRilGJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.126. Megjegyz\303\251s.</Font></Text-field> b:='b'; f:=x/(x-b)^2; convert(f,parfrac,x); SSJiRzYi KiZJInhHNiIiIiIpLCZGI0YlSSJiR0YkISIiIiIjRik= LCYqJkkiYkc2IiIiIiksJkkieEdGJUYmRiQhIiIiIiNGKkYmKiRGKEYqRiY= f:=(4*x^3-6*x^2-2)/(x^4-2*x^3-2*x+4); convert(f,parfrac,x); KiYsKComIiIlIiIiKUkieEc2IiIiJEYmRiYqJiIiJ0YmKUYoIiIjRiYhIiJGLkYvRiYsKiokKUYoRiVGJkYmKiZGLkYmRidGJkYvKiZGLkYmRihGJkYvRiVGJkYv LCYqKCIiJCIiIilJInhHNiIiIiNGJSwmKiQpRidGJEYlRiVGKSEiIkYtRiUqJCwmRidGJUYpRi1GLUYl convert(f,parfrac,x,2^(1/3)); LCgqKiwmIiIiISIiKiQpIiIjI0YpIiIkRiVGJUYlKUYpI0YlRitGJSwmSSJ4RzYiRiUqJEYsRiVGJkYmLCZGKUYmRjFGJUYmRiYqKiIiJ0YlLCZGL0YlRilGJkYmLCgiIiVGJSomRilGJUYsRiVGJUYnRiVGJkYyRiZGJiomLCYqJkYpRiVGL0YlRiZGMUYmRiUsKCokKUYvRilGJUYmKiZGL0YlRixGJUYmRidGJkYmRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.127. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.128. T\303\266bbhat\303\241rozatlan\303\272 polinomok.</Font></Text-field> p:=3*x^3*y^2+4*x^3*y+7*x^2*y^2*z+9; indets(p); coeff(p,x^3); coeff(p,y^2); degree(p,x); degree(p,{x,z}); degree(p); LCoqKCIiJCIiIilJInhHNiJGJEYlKUkieUdGKCIiI0YlRiUqKCIiJUYlRiZGJUYqRiVGJSoqIiIoRiUpRidGK0YlRilGJUkiekdGKEYlRiUiIipGJQ== PCVJInhHNiJJInpHRiRJInlHRiQ= LCYqJiIiJCIiIilJInlHNiIiIiNGJUYlKiYiIiVGJUYnRiVGJQ== LCYqJiIiJCIiIilJInhHNiJGJEYlRiUqKCIiKEYlKUYnIiIjRiVJInpHRihGJUYl IiIk IiIk IiIm LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*8.3.129. Multiindexek.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.130. Form\303\241lis hatv\303\241nysorok.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.131. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">8.3.132. Megjegyz\303\251s.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.133. Szimmetrikus polinomok.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.134. Szimmetrikus polinomok alapt\303\251tele.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*8.3.135. Newton k\303\251pletei.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->8.3.136. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.137. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.138. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.139. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.140. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*8.3.141. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.142. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok megold<Font encoding="UTF-8">\303\241sokkal.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">8.3.143. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
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