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.
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.
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. 2. 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) — more general patterns this builds on
-
Algebraic number field is a kind of Embedding Prime
The proposed strict upward parent is
prime:embedding.
Hierarchy path (1) — routes to 1 parentless root
- Algebraic number field → Embedding → Representation → Abstraction
Neighborhood in Abstraction Space¶
Algebraic number field sits in a crowded region of the domain-specific corpus (9th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Polynomial Algebra & Field Structure (25 abstractions)
Nearest neighbors
- Golden field — 0.94
- Algebraically closed field — 0.93
- Local class field theory — 0.92
- Class number formula — 0.92
- Polynomial identity ring — 0.92
Computed from structural-signature embeddings · 2026-09-08