[代數] 代數數論

[bineapple (Bineapple)]

這是證明如果 A is an integral domain with finitely many prime ideals,

then A is a Dedekind domain if and only if A is a PID 的一部份

James Milne寫的Algebraic number theory中的Corollary 3.12

他寫

Let p_1, ..., p_m be prime ideals in Dedekind domain A.

Choose an element x_1 in p_1-p_1^2. According Chinese number theorem,

there is an element x in A such that

x=x_1 mod p_1^2, x=1 mod p_i, i is not 1.

Now the ideals p_1 and (x) generate the same ideals in A_(p_i) for all i,

i.e., p_1A_(p_i) = (x)A_(p_i),

where A_(p_i)=A(A-p_i)^(-1).

我最後一句不知道在i=1時為什麼可以成立

有沒有高手能夠幫我說明一下呢?

謝謝!

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◆ From: 126.125.156.37
※ 編輯: bineapple       來自: 126.125.156.37       (06/06 21:13)
推 yusd24      :A如果只有一個prime又是domain不就要是field嗎？
→ yusd24      :所以你應該可以假設 i>1
※ 編輯: bineapple       來自: 126.125.156.37       (06/06 22:12)
→ bineapple   :不好意思 剛才發現有打錯
→ bineapple   :這邊A裡面應該有m個prime ideal
※ 編輯: bineapple       來自: 126.125.156.37       (06/06 22:19)