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.

William Stein 2012-09-24