推 ss1132 :謝謝你 04/13 16:51
※ 引述《ss1132 (景)》之銘言：
: 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)
: 證f(x,y)屬於Q[y]沒有根
: 三次式沒有根就是irreducible
: 然後就卡住了
: 麻煩大家謝謝
簡短提供一些步驟
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].
如有什麼地方有問題請告訴我 .
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◆ From: 59.112.234.165
※ 編輯: keroro321 來自: 59.112.234.165 (04/13 16:13)
※ 編輯: keroro321 來自: 59.112.234.165 (04/13 16:17)