## In a quadratic field, rational primes have at most two irreducible factors

Let $K$ be a quadratic extension of $\mathbb{Q}$ and let $p$ be a rational prime. Prove that, as an algebraic integer in $K$, $p$ has at most two irreducible factors.

Recall that the conjugates of $p$ for $K$ are $p$ itself with multiplicity 2. So the norm of $p$ over $K$ is $p^2$. Since (by Lemma 7.1) the norm of an algebraic integer is a rational integer and the norm is multiplicative, $p$ has at most two irreducible integer factors in $K$.