Algebraic number theory

Algebraic number theory studies arithmetic in number fields, using algebraic integers, ideals, and related structures to generalize properties of ordinary integers.

Number field: A finite field extension of the rational numbers. Its elements provide the arithmetic setting that the subject studies.
Ideal class group: The group of nonzero fractional ideals modulo principal fractional ideals in a number field. Its elements record exactly how far the ring of integers is from unique element factorization.
Unique factorization of ideals: The theorem that every nonzero ideal in a Dedekind domain factors uniquely into prime ideals. It replaces unique factorization of elements, which can fail in rings of integers.
Ernst Kummer: A German mathematician whose work on cyclotomic fields introduced ideal numbers. His response to failed unique factorization anticipated the modern theory of ideals.
Class field theory: The theory describing abelian extensions of number fields through arithmetic invariants of the base field. It is the mature culmination of reciprocity laws and a foundation for broader conjectures.
Algebraic integer: A complex number that satisfies a monic polynomial with integer coefficients. These are the number-field analogues of ordinary integers.
Unit group: The multiplicative group of invertible elements in a ring. Units govern which elements generate the same principal ideal and have a structure described by Dirichlet's theorem.
Quadratic field: A degree-two field extension of the rational numbers. Quadratic fields provide the simplest setting for explicit calculations of units, ideals, and class groups.
Richard Dedekind: A German mathematician who developed ideals and the modern theory of algebraic integers. His ideal theory gave number fields a robust replacement for element factorization.
Artin reciprocity law: A theorem identifying the Galois group of an abelian extension with a quotient of an ideal class group. It captures the central correspondence that general nonabelian extensions still lack.

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