Dirichlet's unit theorem

For a number field with r real embeddings and s pairs of complex embeddings, its ring of integers has unit group isomorphic to a finite group of roots of unity times ℤ^(r+s−1).

Unit group: The group of invertible elements in a ring, with multiplication as its operation. The theorem describes the structure of this group for rings of integers.
Number field: A finite extension of the field of rational numbers. The theorem applies to the integers inside each such extension.
Fundamental unit: A generator of the free part of the unit group of a real quadratic field. A real quadratic field has rank one, so one fundamental unit generates its free part.
Gaussian integers: The ring ℤ[i] of complex numbers with integer real and imaginary parts. Its field has no real embeddings and one complex pair, so the unit rank is zero.
Peter Gustav Lejeune Dirichlet: A German mathematician whose work helped establish analytic number theory and modern number theory. He proved the theorem on the structure of units in number fields.
Ring of integers: The subring of a number field consisting of elements that satisfy monic polynomials with integer coefficients. Its invertible elements are the units whose rank the theorem computes.
Field embedding: An injective homomorphism from one field into another that preserves addition, multiplication, and identity. The theorem counts embeddings of the field into the complex numbers.
Pell's equation: The Diophantine equation x² − Dy² = 1 for a positive nonsquare integer D. Solutions correspond to norm-one units in quadratic rings, often generated from one unit.
Eisenstein integers: The ring ℤ[ω], where ω is a primitive cube root of unity. Its field also has one complex pair and rank-zero units, including six roots of unity.
Algebraic number theory: The study of algebraic extensions of the rationals and arithmetic within them. The theorem is a foundational structural result in this subject.

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