A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
The ideal class group of a number field K measures the failure of unique factorization in its ring of integers. Its order (the class number) quantifies how far the ring is from being a principal ideal domain, and is a central invariant in algebraic number theory.
From your study of the failure of unique factorization, you know that rings like ℤ[√−5] do not enjoy the unique factorization property that makes ℤ so well-behaved. In ℤ[√−5], the number 6 factors as both 2 · 3 and (1 + √−5)(1 − √−5), with none of these four elements being further reducible. The ideal class group is the machinery mathematicians invented to understand and measure exactly this kind of failure.
The key idea is to shift attention from elements to ideals — subsets of the ring closed under addition and under multiplication by any ring element. In ℤ, every ideal is principal: it consists of all multiples of a single generator, written (n). But in ℤ[√−5], some ideals are not generated by a single element. The ideal class group captures this: it is the group of fractional ideals of the ring of integers 𝒪_K, modulo the subgroup of principal ideals. Two ideals are equivalent if their ratio is principal. The group operation is ideal multiplication.
The class number h(K) is the order of this group — the number of equivalence classes. When h(K) = 1, every ideal is principal, and unique factorization holds for elements. In ℤ[√−5], the class number is 2, and the two classes correspond precisely to the two "types" of factorizations of 6: even though 2 · 3 ≠ (1+√−5)(1−√−5) as element factorizations, the ideal factorizations agree — both equal the product of ideal primes (2, 1+√−5) and (2, 1−√−5) and (3, 1+√−5) and (3, 1−√−5). Ideals restore unique factorization even when elements do not have it.
The class group is a finite abelian group, and computing it is a central problem in algebraic number theory. It governs which integers are norms of elements in the ring, which Diophantine equations have solutions, and — historically — whether Fermat's Last Theorem holds for a given prime exponent. Kummer's attempt on Fermat's Last Theorem failed precisely because he initially assumed class number 1 for cyclotomic rings; the correction led him to develop the full theory of ideals. The Minkowski bound gives an effective upper bound on which primes need to be checked to compute the class group, making h(K) algorithmically computable for any number field.