Suppose is a quadratic field. Then is Galois, so for each prime we have . There are exactly three possibilties:
• 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, we will later show that the ramified primes are exactly the ones that divide the discriminant.)
• Inert: , , : The prime is inert in , so is prime. This happens 50% of the time, which is suggested by quadratic reciprocity (but not proved this way), as we will see illustrated below for a particular example.
• Split: , : The prime splits in , in the sense that with . This happens the other 50% of the time.
Suppose, in particular, that , 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 .