Number Fields and Rings of Integers

A number field is a finite extension of the rational numbers, and its ring of integers is the natural arithmetic analogue of the ordinary integers — a Dedekind domain in which ideals, not elements, factor uniquely.

Definition

A number field is a finite-degree field extension of the rational numbers; its ring of integers consists of the elements that are roots of monic polynomials with integer coefficients, forming a Dedekind domain.

Scope

This topic covers algebraic numbers and algebraic integers, number fields and their degree and embeddings, the ring of integers as the integral closure of the integers in the field, integral bases and the field discriminant, the characterization of rings of integers as Dedekind domains, and the unique factorization of nonzero ideals into prime ideals.

Core questions

• Which elements of a number field count as integers, and why do they form a ring?
• What is an integral basis, and how is the discriminant of a number field defined and computed?
• What properties make the ring of integers a Dedekind domain?
• How does unique factorization of ideals replace unique factorization of elements?

Key theories

Ring of integers and integral closure

The algebraic integers in a number field form its ring of integers, the integral closure of the integers in the field; it is a free module of rank equal to the field degree, with an integral basis.

Dedekind domains and factorization of ideals

Rings of integers are Noetherian, integrally closed, of dimension one — that is, Dedekind domains — and in any Dedekind domain every nonzero ideal factors uniquely into prime ideals.

Discriminant

The discriminant of an integral basis is an integer invariant of the field that detects ramified primes and constrains the field via Minkowski's bound and Hermite's finiteness theorem.

Clinical relevance

Rings of integers and their ideal structure are the setting for the number field sieve factorization algorithm and for ideal-lattice cryptography, where the arithmetic of an integer ring is the source of both hard problems and efficient operations.

History

Kummer worked with cyclotomic integers and ideal numbers in the 1840s. Dedekind, in supplements to Dirichlet's lectures from the 1870s, defined the ring of integers and the modern notion of an ideal, proving unique factorization of ideals and founding the abstract theory.

Key figures

• Richard Dedekind
• Leopold Kronecker
• Ernst Kummer

Seminal works

• marcus2018

Frequently asked questions

Is the ring of integers always a unique factorization domain?

No. Elements need not factor uniquely, but the ring is always a Dedekind domain, so ideals do; the ring is a unique factorization domain exactly when its class number is one.

What does the discriminant tell you?

The field discriminant is an integer invariant whose prime divisors are exactly the primes that ramify in the field, and its size bounds how complicated the field can be.