Ring of integers

The ring of integers of a number field consists of all elements satisfying monic polynomials with integer coefficients. It is the field’s canonical domain for studying divisibility and prime ideals.

Algebraic number field: A finite field extension of the rational numbers. The ring of integers is defined separately for each such field.
Dedekind domain: A Noetherian integrally closed domain in which every nonzero prime ideal is maximal. Every ring of integers is a Dedekind domain, enabling unique factorization of ideals.
Gaussian integers: The ring of numbers a+bi with integers a and b, where i²=−1. They form the ring of integers of the quadratic field of rational multiples of i.
Ernst Kummer: A nineteenth-century German mathematician who developed ideal numbers to study arithmetic in cyclotomic fields. His ideal numbers anticipated the ideals used to restore factorization in rings of integers.
Algebraic integer: A complex number that satisfies a monic polynomial with integer coefficients. The ring consists exactly of the algebraic integers lying in the chosen field.
Prime ideal: A proper ideal whose quotient ring is an integral domain. Prime ideals replace prime elements when elements do not factor uniquely.
Eisenstein integers: The ring generated over the integers by a primitive cube root of unity. They illustrate a quadratic ring of integers with distinctive factorization properties.
Richard Dedekind: A German mathematician who founded modern algebraic number theory and introduced ideals in their modern form. His work established the structural theory of rings of integers and ideal factorization.
Integral element: An element of a ring extension that satisfies a monic polynomial over the base ring. Algebraic integers are integral over the ordinary integers.
Ideal class group: The group of nonzero fractional ideals modulo principal fractional ideals. It measures the failure of elements in the ring to have unique factorization.

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