[代數] ring
自修代數有問題不清楚，特上來求教各位大大
Consider the ring Z[√-3]
(1) Prove that 5 is an irreducible element in Z[√-3].
(2) Prove that 5 is not a prime element in Z[√-3].
(3) Is Z[√-3] a unique factorization domain? Explain your answer.
(4) Is Z[√-3]/(5) a field? Explain your answer.
(1)
Let R=Z[√-3]. The set of units in R={1,-1, i, -i}
Suppose there are a+b√-3, c+d√-3 in R such that (a+b√-3)(c+d√-3)=5.
Take norm we obtain (a^2+3b^3)(c^2+3d^2)=25.
If a^2+3b^2=1, then a+b√-3 is a unit. If a^2+3b^2=25, then c+d√-3 is a unit.
And no integers satisfy a^2+3b^2=5. Hence 5 is irrecudcible.
請問這樣寫可以嗎?
(2) R is a ring. a is prime in R if a|bc implies a|b or a|c. (b, c in R)
若要證 5 is not prime, 那我應該找到一組 a+b√-3, c+d√-3
使得 5|(a+b√-3)(c+d√-3) 但 5 不整除 a+b√-3 或 c+d√-3.
但是我找不到這樣的a+b√-3, c+d√-3. 所以 5 is prime in Z[√-3]?
應該怎麼寫好呢?
(3) 我有看到一個定理: If R is a UFD, then a in R is an irreducible element
if and only if a is a prime element. 應該是用這個證吧? 但該怎麼寫呢?
(4) 因為 Z[√-3] 是 a commutative ring with 1. 應該是利用以下定理:
R/M is a field if and only if M is a maximal ideal in R.
所以目標是要證 (5) 是一個 maximal ideal嘍?
感謝各位大大賜教
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