# Prove ideals generated by primes split as distinct or equal prime ideals?

I'm new to algebraic number theory and have the following questions: consider the quadratic field $$K = \mathbb{Q}(\sqrt{2})$$. Let $$p \in \mathbb{Z}$$ be a prime number, and $$(p)$$ be the ideal generated by $$p$$ in the ring $$O_K$$ of algebraic integers in $$K$$.

A) Prove that if $$p = 2$$ then $$(p)$$ splits as the product of two equal prime ideals.

B) Prove that if $$p$$ is an odd prime and $$\frac{p^2−1}{8}$$ is odd then $$(p)$$ is a prime ideal.

C) Prove that if $$p$$ is an odd prime and $$\frac{p^2−1}{8}$$ is even then $$(p)$$ splits as the product of two distinct prime ideals.

My thoughts so far: I know we have an isomorphism $$O_K \simeq \mathbb{Z}[x]/(x^2 −2)$$, which I think should help but I'm not sure how.

Let $$R = \mathcal{O}_{K}$$. As you note, $$R = \mathbb{Z}[\sqrt{2}] \cong \mathbb{Z}[X]/\langle X^{2}-2 \rangle$$. Note that for any prime $$p$$, $$R/pR \cong \mathbb{Z}[X]/\langle p, X^{2}-2 \rangle \cong (\mathbb{Z}/p\mathbb{Z})[X]/\langle X^{2}-2 \rangle$$. (In the last isomorphism, I am being a bit sloppy and identifying $$X^{2}-2$$ with its residue class in $$(\mathbb{Z}/p\mathbb{Z})[X]$$.)
The factorization of $$p$$ into a product of prime ideals of $$R$$ can consequently be understood in terms of how the polynomial $$X^{2}-2$$ factors over $$\mathbb{Z}/p\mathbb{Z}$$. There is a general theorem at work here, namely Dedekind's theorem, but you can attack things quite directly. (I would personally view this problem as a nice way to see how Dedekind's theorem works in a simple case.) Here are some hints for each part:
(A) See if you can prove that $$\sqrt{2}R$$ is a prime ideal; it shouldn't be hard from there to show that $$2R = (\sqrt{2}R)^{2}$$. (Hint: $$R/\sqrt{2}R \cong \mathbb{Z}[X]/\langle X, X^{2}-2\rangle \cong \ldots$$). This reflects the fact that $$R/2R \cong (\mathbb{Z}/2\mathbb{Z})[X]/\langle X^{2} \rangle$$.
(B), (C): The conditions on $$p$$ here control whether $$2$$ is a quadratic residue mod $$p$$, i.e. whether $$X^{2}-2$$ splits over $$\mathbb{Z}/p\mathbb{Z}$$. You should try to show that in (B), $$2$$ is NOT a quadratic residue mod $$p$$, and so $$X^{2}-2$$ is irreducible over $$\mathbb{Z}/p\mathbb{Z}$$, in which case $$pR$$ is prime. Likewise, in (C), $$2$$ IS a quadratic residue mod $$p$$. Since $$p$$ is odd, $$X^{2}-2$$ splits into distinct factors over $$\mathbb{Z}/p\mathbb{Z}$$. Try to use this factorization (and maybe the Chinese Remainder Theorem) to construct the prime factorization of $$pR$$.
• Thanks for your response. I can follow most of it, but am not sure on a couple of points: for A, I don't know how to prove that $\sqrt{2}R$ is a prime ideal. For C, I'm not sure how to find the distinct factors of $X^2-2$ to construct the prime factorisation of $pR$. – user766821 Apr 3 '20 at 20:57