Ideals restore unique factorisation — but they also let us measure exactly how badly ordinary numbers failed at it. That measurement is the ideal class group, one of the central invariants of a number field.

Principal vs. non-principal ideals

Some ideals are just "the multiples of a single number" — these are called principal. When every ideal is principal, ideals and numbers coincide and ordinary unique factorisation holds. Failure of unique factorisation is precisely the existence of non-principal ideals — ideals that no single number generates.

The class group

Group the ideals into classes, declaring two equivalent if they differ by a principal ideal. These classes form a finite group under multiplication — the ideal class group — and its size is the class number h.

The Gaussian integers have h = 1 (factorisation is fine); but \mathbb{Z}[\sqrt{-5}] has h = 2 — the smallest possible failure, exactly the one we saw with 6.

A deep, still-mysterious invariant

The class number is finite but erratic, and computing it is hard. Gauss conjectured there are only finitely many imaginary quadratic fields with h = 1 (there are exactly nine), while whether infinitely many real quadratic fields have h = 1 is still open. Class numbers connect to L-functions through Dirichlet's class number formula, weaving the algebraic and analytic threads of this course together at its very end.