## Prime Decomposition

It makes sense to ask what the relationship is between a prime (integral) ideal P in some number field K, and a prime integer in Z. For a start, we can look at the intersection of P with Z; it’s straightforwardly a prime ideal, since rules like “ab is in P implies a or b is in P” are true by restriction to Z. It’s also non-zero, since if a in P is nonzero, then N(a) in P and Z is nonzero.

Since the intersection of P with Z is a prime ideal, we can denote it by (p). We then say that P lies over (p). Since (p) is an ideal of K, too, we can look at it as an ideal that’s contained in P, i.e. is divisible by P. So the set of all prime ideals of K that lie over (p) is just the set of all prime ideals of K that occur in the unique factorization of the ideal (p).

For a concrete example, let K = Q(i), O(K) = Z[i]. O(K) is a PID, as I showed ages ago. If p = 1 mod 4, then p = a^2 + b^2 for some integers a and b, so (p) = (a + bi)(abi), and we say that p splits; in a more general setting, p splits completely if it factors into ideals that have norm p. If p = 3 mod 4, then N(p) = p^2, so any proper factor will have norm p, which is impossible since it would imply a^2 + b^2 = p; in that case, the ideal (p) is prime so we say p remains prime, or is inert. If p = 2, then (p) = (1 + i)^2 and p is called ramified; in fact (p) = P^n and N(P) = p so P totally ramifies.

At this point it’s not clear what the difference between ramifying and splitting is, but in more advanced parts of number theory, ramification is a fairly pathological thing, which unfortunately happens for at least some primes in a number field (more precisely, p ramifies in K iff p divides the discriminant of K).

To see how prime decomposition is useful, note that if P is any prime ideal of Z[i] then it lies over some p. If p = 3 mod 4, then P = (p); otherwise, P is generated by a single element of norm exactly p. This is nontrivial, because it’s not clear that just like 3 is an irreducible element of Z[i] with composite norm, so could there be an element of the form a + bi where a and b are nonzero that is irreducible but has composite norm.