作者keroro321 (日夕)
看板Math
标题Re: [代数] irreducible
时间Wed Apr 13 16:08:35 2011
※ 引述《ss1132 (景)》之铭言：
: Prove that
: f(x,y)=xy^3+x^2y^2-x^5y+x^2+1
: is an irreducible polynomial in /R[x,y]
: 我的想法是令Q=/R(x)
: 证f(x,y)属於Q[y]没有根
: 三次式没有根就是irreducible
: 然後就卡住了
: 麻烦大家谢谢
简短提供一些步骤
Let |R = |R[x] , F(y) = f(x,y) is a primitive polynomial in |R [y].
1 1
if F(y) = ( B1 y + C1 )( A2 y^2 + B2 y + C2 )
where A1,B1,B2,C1,C2 all are elements of |R[x] .
┌
│ x^2 + 1 = C1 * C2
(*) ─│ x = B1 * A2
│ x^2 = B1*B2 + C1*A2
│-x^5 = B1*C2 + C1*B2
└
First , we observe that
C1≠0 , C2≠0, B1≠0 , A2≠0
deg ( B1*C2 ) ≦ 3 .
=> B2≠0 and deg ( C1*B2 ) = 5
you find that A1,B1,B2,C1,C2 must satisfy
deg(C2) = deg(B1) = 0 , deg(B2) = 3 , deg(C1) = 2 .
After carefully examining it (Do it yourself !!)
, you will conclude that
"F(y) cannot be factored into two polynomials of
lower degree in |R [y]."
1
, so f(x,y) is irreducible in |R[x,y].
如有什麽地方有问题请告诉我 .
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※ 编辑: keroro321 来自: 59.112.234.165 (04/13 16:13)
※ 编辑: keroro321 来自: 59.112.234.165 (04/13 16:17)
1F：推 ss1132 :谢谢你 04/13 16:51