Suppose is a quadratic field. Then is Galois, so for each prime we have . There are exactly three possibilities:
• Ramified: , : The prime ramifies in , so . There are only finitely many such primes, since if is the minimal polynomial of a generator for , then ramifies if and only if has a multiple root modulo . However, has a multiple root modulo if and only if divides the discriminant of , which is nonzero because is irreducible over . (This argument shows there are only finitely many ramified primes in any number field. In fact, the ramified primes are exactly the ones that divide the discriminant.)
• Inert: , , : The prime is inert in , so is prime. It is a nontrivial theorem that this happens half of the time, as we will see illustrated below for a particular example.
• Split: , : The prime splits in , in the sense that with . This happens the other half of the time.
For example, let , so , where . Then is ramified, since . More generally, the order has index in , so for any prime we can determine the factorization of in by finding the factorization of the polynomial . The polynomial splits as a product of two distinct factors in if and only if and . For this is the case if and only if is a square in , i.e., if , where is if is a square mod and if is not. By quadratic reciprocity, Thus whether splits or is inert in is determined by the residue class of modulo . It is a theorem of Dirichlet, which was massively generalized by Chebotarev, that half the time and the other half the time.