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.

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