Part 10 of 13. Unique factorization fails in . Kummer and Dedekind restored it by factoring ideals instead of numbers.
The ring consists of the numbers with integers. Its norm
is multiplicative, so a factorization of gives one of the integer . Plotted at , the elements of norm are the lattice points on the circle of radius .
Figure 1. The lattice at true scale, with the circle of norm . The strip runs over ; hollow red dots are norms no element has. Click a point or the strip, or use the arrow keys.
The units, of norm 1, are . The empty circles at 2 and 3 settle irreducibility, because a proper factor of 2, 3 or would need norm 2 or 3. So all four atoms are irreducible, no two differ by a unit, and
are two different factorizations.
Ernst Kummer met this failure in 1847 in the cyclotomic integers , working on higher reciprocity laws, and patched it with "ideal numbers" outside the ring; Fermat's Last Theorem came after, not before (Edwards, 1975). Dedekind in 1871 replaced them with ideals, subsets of the ring itself.
The ideal is the set of sums with each in the ring. Here a nonzero ideal is a sublattice, its index is its norm , and a product of ideals is generated by products of generators. Three ideals do the work for 6:
Figure 2. Filled points belong to the ideal; the shaded cell has area in units of the ring's cell. The product is computed exactly (Hermite normal form) and is principal exactly when it contains an element whose norm equals its index.
The index multiplies, , for every pair. No element has norm 2, so has no single generator, yet . The other pairings give , and , so both factorizations of 6 group the same four prime ideals:
In a Dedekind domain such as this one, every nonzero ideal is a product of prime ideals in exactly one way. Figure 3 finds every factorization of an element by a divisor search on norms, then splits each factor into prime ideals, with over 7.
Figure 3. Every factorization into irreducibles, up to order and sign. Tiles show the prime ideals in each factor, in that factor's colour. The elements change from row to row; the tiles do not.
The number of factors holds still too. Carlitz showed in 1960 that in a ring of integers this happens exactly when the class group, below, has at most two elements.
The class group is the group of nonzero fractional ideals modulo the principal ones. It is trivial exactly when factorization of elements is unique, so its size measures the failure. A prime ideal of norm is principal exactly when some element has norm .
Figure 4. Primes below 200 by : green if the ideals over are principal, violet if not, hollow if stays prime. Pick two primes to multiply their ideals; the second line multiplies every pair.
The colour depends only on (the classical rule: exactly when ), and every pair obeys the table of . That is evidence; the proof is Minkowski's bound, which puts an ideal of norm at most in every class, leaving only and .
The class number of equals the number of reduced forms of discriminant . For they are the norm and , the norm on .
Figure 5. Class numbers of , squarefree , (where it is the full ring of integers of ). Green: class number 1. Click a bar or use the arrow keys for its forms.
Dedekind proved unique factorization into prime ideals for the ring of integers of every number field. Noether's Idealtheorie in Ringbereichen (1921) worked in any commutative ring with the ascending chain condition, where every ideal is an intersection of finitely many primary ideals. In 1927 she showed Dedekind's theorem holds exactly in the Noetherian, integrally closed rings whose nonzero primes are maximal: the Dedekind domains. The next part states the chain condition.