Noetherian Rings - R Y Sharp - Chapter 8 - exercise 8.5

[Math Amateur]

I am reading R. Y. Sharp: Steps in Commutative Algebra, Chapter 5 - Commutative Noetherian Rings

Exercise 8.5 on page 147 reads as follows:

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8,5 Exercise.

Show that the subring $$\mathbb{Z} [ \sqrt{-5} ]$$ of the field $$\mathbb{C}$$ is Noetherian.

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Can someone please help me get started on this problem and give me a solution strategy.

Peter

[Note: This has also been posted on MHF]

 

[mathbalarka]

From Hilbert's basis theorem, we know that since $\Bbb Z$ is Noetherian, $\Bbb Z[x]$ is also Noetherian. There is a canonical isomorphism $\Bbb Z[x]/(x^2+5) \cong \Bbb Z[\sqrt{-5}]$ given by $x \mapsto \sqrt{-5}$. As Noetherianty is invariant under taking quotients, $\Bbb Z[x]/(x^2 + 5)$ is Noetherian, hence $\Bbb Z[\sqrt{-5}]$ is also Noetherian $\blacksquare$