is a quadratic field. Then is Galois, so for each prime
. There are exactly three possibilties:
Suppose, in particular, that
. 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
. 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
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
- 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
. This happens the other
50% of the time.