Ideals repaired unique factorisation by promoting "the multiples of a number" into first-class objects of their own. But fixing something invites a sharper question: exactly how badly did the ordinary numbers fail in the first place? Was \mathbb{Z}[\sqrt{-5}] a little bit broken, or catastrophically broken? Is there a way to attach a single number to a number field that says precisely how far it sits from ordinary unique factorisation?

There is. It's called the class number, written h, and it comes from a structure called the ideal class group. When h = 1 there is no failure at all — numbers factor as nicely as ordinary integers do. Every value of h bigger than 1 is a real, measurable amount of breakage, and the ideal class group tells you exactly how that breakage is structured.

Think of it like a "leftover meter". Ideals restore perfect factorisation at the level of ideals, so anything left unaccounted for at the level of ordinary numbers must be some kind of leftover residue — numbers that keep clashing no matter how you group their factors. The ideal class group collects every distinct flavour of that leftover residue into one tidy, finite, countable object, and its size is the class number.

Principal vs. non-principal ideals

Some ideals are just "the multiples of a single number" — these are called principal, written (\alpha) for the generator \alpha. When every ideal in a ring of integers is principal, ideals and numbers coincide, and ordinary unique factorisation of numbers holds exactly as it does in \mathbb{Z}.

Failure of unique factorisation is precisely the existence of non-principal ideals — ideals that behave perfectly well as ideals (Dedekind's theorem still factors them uniquely into primes) but that no single element of the ring can generate. The ideal class group is built by sorting every ideal into "how far from principal" it is.

Two ideals count as the same "how far" if one turns into the other after multiplying by a principal ideal — dividing out the part that was never broken in the first place. What's left, once you quotient away all the principal noise, is a clean record of exactly the non-principal behaviour the ring exhibits.

The class group

Group the (fractional) ideals of a number field into classes, declaring two ideals equivalent when they differ by a principal factor. Remarkably, these classes are always finite in number, and they carry their own multiplication (more on that below) — the resulting structure is the ideal class group, and its size is the class number h.

Worked example: \mathbb{Q} itself has class number 1

Start with the simplest number field of all: the ordinary rationals \mathbb{Q}, whose ring of integers is just \mathbb{Z}. Every ideal of \mathbb{Z} has the form (n) = n\mathbb{Z} for some integer n — a single generator, every time. There is no ideal that fails to be generated by one number, so there are no extra classes to collect.

The class group of \mathbb{Q} is trivial, and h = 1. This is exactly what the fundamental theorem of arithmetic already told us in elementary form: ordinary integers factor uniquely into primes, full stop, no repair ever needed. The Gaussian integers behave the same way — their class number is also 1.

Worked example: \mathbb{Q}(\sqrt{-5}) has class number 2

Now the field where things actually broke. Recall the clash 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}), which refines uniquely into prime ideals as (2) = \mathfrak{p}_2^{\,2} and (3) = \mathfrak{p}_3\,\mathfrak{p}_3'. The ideal \mathfrak{p}_2 = (2,\, 1+\sqrt{-5}) turns out to be non-principal — no single element of \mathbb{Z}[\sqrt{-5}] generates it.

But watch what happens when you square it: \mathfrak{p}_2^{\,2} = (2), a perfectly principal ideal! So the non-principal class, multiplied by itself, lands you right back at the identity (the class of principal ideals). The class group of \mathbb{Q}(\sqrt{-5}) has exactly two elements — the principal class and the class of \mathfrak{p}_2 — so it is the cyclic group \mathbb{Z}/2\mathbb{Z}, and h = 2. That "2" is not a coincidence: it is the precise number that quantifies the earlier failure of 6's factorisation as exactly one extra layer of non-uniqueness — no more, no less.

Worked example: a "third layer" in \mathbb{Q}(\sqrt{-23})

A class number of 2 is the smallest possible failure — but it isn't the only shape failure can take. In \mathbb{Q}(\sqrt{-23}), the ring of integers has class number h = 3. There is a non-principal ideal \mathfrak{a} whose class does not return to the identity after squaring — you need \mathfrak{a}^3 (not \mathfrak{a}^2) to land back on a principal ideal. The class group is the cyclic group \mathbb{Z}/3\mathbb{Z}: three distinct classes, arranged in a loop of length three rather than a simple on/off switch of length two.

This is the general pattern: the class number isn't just "how much" failure there is, it also encodes the shape of the failure — a cyclic group of order 2, of order 3, or (for more complicated fields) a product of several cyclic pieces at once, exactly like any other finite abelian group can be built.

A genuine group, born from arithmetic

The real payoff of sorting ideals into classes is that the classes can be multiplied. Given two classes [I] and [J], define [I]\cdot[J] = [IJ] — multiply representative ideals, then take the class of the product. This is well-defined (it doesn't matter which representatives you pick), the class of principal ideals acts as the identity [\mathcal{O}], and — because the class group is finite — every class has an inverse.

That makes the ideal class group a bona-fide finite abelian group, sitting at the crossroads of number theory and abstract algebra: an arithmetic question ("how badly does factorisation fail?") gets answered by algebraic structure ("what group is this?"). For \mathbb{Q}(\sqrt{-5}) that group is the humble \mathbb{Z}/2\mathbb{Z}, but other number fields have class groups that are large, or built from several cyclic pieces at once.

Where do the inverses come from? A general ideal I in a ring of integers can always be "undone" by a fractional ideal I^{-1} — an allowed generalisation of an ideal that admits denominators — with I \cdot I^{-1} = \mathcal{O}, the whole ring, which is exactly the principal (identity) class. Every ideal has such an inverse, so every class does too, and the group axioms are satisfied completely: closure, an identity, inverses, and associativity inherited from ordinary multiplication.

A deep, still-mysterious invariant

The class number is always finite, but wildly unpredictable as the field changes — a beautifully erratic sequence that number theorists have studied for two centuries and still don't fully understand. Dirichlet's class number formula ties h directly to the value of an L-function at a special point, weaving the algebraic thread of this course (ideals, class groups) together with its analytic thread (L-functions, zeta functions) at the very end. Few single numbers in mathematics carry so much of the field's structure in them.

The idea of "measuring failure with a group" is bigger than number fields, too. A close cousin of the class number — attached not to a number field but to an elliptic curve — sits at the heart of the Birch and Swinnerton-Dyer conjecture, one of the seven Clay Millennium Prize Problems, still unsolved and worth a million dollars to whoever settles it. The same instinct that led Kummer and Dedekind to count "how broken" a ring of integers is turns out to be one of the deepest running themes in all of modern mathematics.