Let be a quadratic extension of and let be a rational prime. Prove that, as an algebraic integer in , has at most two irreducible factors.

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

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