Ideal classes. We are nearly ready to justify the assertion that each ideal in K = R(θ) is the totality of integers in K which are divisible (in the extended sense) by some integer which is not necessarily in K. Our proof will rest on the notion of ideal class.

Two ideals A and B in K are equivalent, written A ∼ B, if there are two non-zero integers α and β in K such that

(α)A = (β)B.

The simplest properties of this equivalence relation are the following:

1. (i) A∼ A;

2. (ii) A ∼ B if and only if B ∼ A;

3. (iii) if A ∼ B and B ∼ C, then A ∼ C;

4. (iv) all non-zero principal ideals are equivalent.

The totality of ideals in K equivalent to a fixed ideal A ≠ (0) is said to constitute a class. The number of classes (which we shall soon show to be finite) is called the class-number h of K. If the class-number is 1 then all non-zero ideals are equivalent to (1) and so are all principal. From Theorem 9.4 it follows that a field has unique factorization of integers into prime integers if and only if its class-number is 1.

Lemma 10.1. Let A, B, C be ideals in K and A ≠ (0). Then AB ∼ AC if and only if B ∼ C.