Algebraic number field¶

A finite-degree field extension of the rational numbers, carrying arithmetic through its ring of integers, embeddings, ideals, norms, traces, units, and completions.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3249
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Algebraic Number Theory, Number Fields → Mathematics
Aliases: Number field

Core Idea¶

An algebraic number field is a field extension K/Q of finite degree, equivalently a field generated over Q by finitely many algebraic numbers and, by the primitive element theorem in characteristic zero, by one algebraic number. A defining polynomial adjoins an algebraic element, reducing powers to a finite basis. Integral closure produces the ring of integers; embeddings, ideals, valuations, and completions expose arithmetic unavailable over Q alone. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

Fractions Plus a Few New Numbers

Start with all the fractions, like one-half and three-quarters. Some numbers, like the one that times itself makes 2, aren't fractions, but they solve a simple equation. If you add a few numbers like that and include everything you can make by adding, subtracting, multiplying, and dividing, you get an algebraic number field.

Fractions Plus a Special Number

The rational numbers are all the fractions. Some numbers, like the square root of 2, aren't fractions, but they solve simple equations with fraction coefficients; those are called algebraic numbers. An algebraic number field is what you get when you add a few algebraic numbers to the fractions and include every result of adding, subtracting, multiplying, and dividing. Because the new numbers solve equations, their powers can be reduced, so every number in the field can be written using only a limited number of building blocks. It turns out you can always get the same field by adding just one well-chosen algebraic number. Mathematicians study these fields to understand whole numbers better.

Finite Extension of the Rationals

An algebraic number field is a field K that contains the rational numbers Q and has finite dimension as a vector space over Q, meaning every element can be written as a Q-combination of a finite basis. Equivalently, it is generated over Q by finitely many algebraic numbers, and by the primitive element theorem, a single algebraic number α is enough, so K = Q(α). The polynomial that α satisfies lets you rewrite high powers of α in terms of lower ones, which is why the dimension is finite. Inside K, the elements that satisfy monic polynomials with integer coefficients form the ring of integers, which plays the role that ordinary integers play inside Q. Studying its ideals, the different ways K embeds into the complex numbers, and related tools reveals arithmetic, such as how primes factor, that you cannot see in Q alone. A field that is infinite-dimensional over Q, such as the real numbers, is not a number field.

An algebraic number field is a finite-degree field extension K/Q, equivalently a field generated over Q by finitely many algebraic numbers, and, by the primitive element theorem in characteristic zero, by a single algebraic number α, so K = Q(α) with degree equal to that of α's minimal polynomial. The minimal polynomial reduces higher powers of α, yielding the finite Q-basis 1, α, ..., α^{d−1}. The integral closure of Z in K is the ring of integers O_K, the arithmetic analogue of Z. Embeddings of K into C, ideals of O_K, valuations, and completions at places expose arithmetic structure, such as factorization of primes and local behavior, that is invisible over Q. The constitutive condition is that K is a field containing an identified copy of Q with finite Q-dimension; infinite algebraic extensions, rings that are not fields, or fields over a different base do not qualify. The concept is the object of algebraic number theory, but its identity is this finite enlargement of Q together with the arithmetic induced by its integers and places.

Scope of Application¶

Algebraic number field belongs to algebraic number theory and is useful where the analyst can specify a field K containing Q, its Q-vector-space structure, and a finite extension degree [K:Q], then evaluate K is a field containing an identified copy of Q and has finite dimension as a Q-vector space. The scope is broad within that domain but bounded by the need for K is a field containing an identified copy of Q and has finite dimension as a Q-vector space. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making K is a field containing an identified copy of Q and has finite dimension as a Q-vector space the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Algebraic number field can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Algebraic number field. Algebraic number field compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: a field K containing Q, its Q-vector-space structure, and a finite extension degree [K:Q]. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express K is a field containing an identified copy of Q and has finite dimension as a Q-vector space independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of algebraic number theory because they reuse a field K containing Q, its Q-vector-space structure, and a finite extension degree [K:Q], A defining polynomial adjoins an algebraic element, reducing powers to a finite basis. Integral closure produces the ring of integers; embeddings, ideals, valuations, and completions expose arithmetic unavailable over Q alone., and type the carrier, state every parameter and convention in the definition, test that K is a field containing an identified copy of Q and has finite dimension as a Q-vector space, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Algebraic number field Domain-specific

Parents (1) — outgoing typed relations

Algebraic number field is a kind of Embedding Prime

A number field contains an embedded copy of Q inside a finite-dimensional field structure.

Children (1) — abstractions linked here by typed relations

Arithmetic Group Domain-specific presupposes Algebraic number field

The construction requires a number field, including Q, and its ring of integers.

Hierarchy path (1) — routes to 1 parentless root

Algebraic number field → is a kind of Embedding → is a kind of Representation → presupposes Abstraction

Neighborhood in Abstraction Space¶

Algebraic number field sits in a crowded region of the domain-specific corpus (6^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Field Theory & Polynomial Structures (26 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25