作者Frank000 (Frank000)
看板Math
标题Re: [代数] 请教两题代数
时间Fri Mar 18 14:06:45 2011
※ 引述《loribank (小萝莉银行)》之铭言：
: 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.
: 感谢帮忙^^
1.很怪，我觉得5在Z[sqrt(-3)]里是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，请其他版友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的因数
(1+sqrt(-19))(1-sqrt(-19))=20
(1+sqrt(-19))(a+b*sqrt(-19))=4or5两边取绝对值不难发现矛盾。
(2)no
1+sqrt(-19)irreducible(假设能分解，两边取绝对值应该可以导出矛盾)
(3)no
Z is PID, Z[x] is not(ideal generated by 2 and x is not principal)
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◆ From: 108.65.2.220
1F：推 ntnusliver :1.同意 03/18 14:59
2F：→ ntnusliver :用norm 可以证得5是prime 03/18 15:01