Re: [代數] 請教兩題代數
※ 引述《loribank (小蘿莉銀行)》之銘言：
: 1. Consider the ring Z[sqrt(-3)]
: (1) Prove 5 is an irreducible element in Z[sqrt(-3)]
: (2) Prove 5 is not a prime element in Z[sqrt(-3)]
: (3) Is Z[sqrt(-3)] ia UFD?
: (4) Is Z[sqrt(-3)]/<5> a field?
: 2. Prove or disprove
: (1) 1+sqrt(-19) is a prime element in Z[1+sqrt(-19)]
: (2) Z[1+sqrt(-19)] is UFD
: (3) If R is PID, then R[x] is a PID.
: 感謝幫忙^^
1.很怪，我覺得5在Z[sqrt(-3)]裡是prime，用excel暴力找也找不到
5|x*y(x,y in Z[sqrt(-3)]) 但5不整除其中之一，請其他版友check。
2.Z[1+sqrt(-19)]=Z[sqrt(-19)]
(1)no 1+sqrt(-19)|5*4=20 but 1+sqrt(-19)不是4或5的因數
(1+sqrt(-19))(1-sqrt(-19))=20
(1+sqrt(-19))(a+b*sqrt(-19))=4or5兩邊取絕對值不難發現矛盾。
(2)no
1+sqrt(-19)irreducible(假設能分解，兩邊取絕對值應該可以導出矛盾)
(3)no
Z is PID, Z[x] is not(ideal generated by 2 and x is not principal)
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