norm of an ideal

We need a way to measure the “size” of an ideal — bigger ideals carve out larger chunks of the ring, and we want a number that captures this. For the integers, the ideal (n) chops Z into exactly |n| residue classes, so its size is |n|. The norm of an ideal generalizes this counting to any ring of integers, and it behaves multiplicatively, just like absolute value does.

For a nonzero ideal I of the ring of integers O_K, the norm N(I) is the cardinality of the quotient ring O_K / I, which is always finite. The key property is multiplicativity: N(I J) = N(I) N(J) for all nonzero ideals, which follows from unique factorization into primes plus the fact that N(P^k) = N(P)^k. For a principal ideal, N((a)) equals the absolute value of the field norm of a.

The ideal norm is the workhorse behind finiteness theorems. There are only finitely many ideals of any given norm, which makes the Minkowski-bound argument possible: bounding the norm bounds the search. For a prime P lying over a rational prime p with residue degree f, the norm is N(P) = p^f, tying the norm directly to the splitting data.