Norm group

The subgroup of a local field’s multiplicative group consisting of field norms from a finite abelian extension.

Version

v1 · 2026-09-08 · History

Domain-specific #

5803

Origin domain

local class field theory

Subdomain

local class field theory

Core Idea

For L over K the image N(L-times) is an open finite-index subgroup of K-times, and local reciprocity classifies finite abelian extensions through these norm groups. Multiplying Galois conjugates maps each nonzero extension element to the base field; the image closes under multiplication and its quotient records the extension’s abelian reciprocity data. 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

Norm group belongs to local class field theory and is useful where the analyst can specify the typed local class field theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the nonarchimedean local field K, finite abelian extension L and degree, multiplicative groups, field norm formula, image subgroup, openness and index, reciprocity map and extension-subgroup correspondence are explicit. The scope is broad within that domain but bounded by the need for the nonarchimedean local field K, finite abelian extension L and degree, multiplicative groups, field norm formula, image subgroup, openness and index, reciprocity map and extension-subgroup correspondence are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the nonarchimedean local field K, finite abelian extension L and degree, multiplicative groups, field norm formula, image subgroup, openness and index, reciprocity map and extension-subgroup correspondence are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

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 Norm group. Norm group 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

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed local class field theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the nonarchimedean local field K, finite abelian extension L and degree, multiplicative groups, field norm formula, image subgroup, openness and index, reciprocity map and extension-subgroup correspondence are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of local class field theory because they reuse the typed local class field theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Multiplying Galois conjugates maps each nonzero extension element to the base field; the image closes under multiplication and its quotient records the extension’s abelian reciprocity data., and type the carrier, state every parameter and convention in the definition, test that the nonarchimedean local field K, finite abelian extension L and degree, multiplicative groups, field norm formula, image subgroup, openness and index, reciprocity map and extension-subgroup correspondence are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Norm group Domain-specific

Parents (1) — more general patterns this builds on

• Norm group is a kind of Group Prime

  The proposed strict upward parent is prime:group.

Hierarchy paths (5) — routes to 5 parentless roots

• Norm group → Group → Monoid → Semigroup → Set and Membership

Show alternative paths (4)

Neighborhood in Abstraction Space

Norm group sits in a crowded region of the domain-specific corpus (28^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Field Extensions & Algebraic Closure (8 abstractions)

Nearest neighbors

• Local class field theory — 0.94
• Restricted representation — 0.91
• Separable polynomial — 0.90
• Minimal polynomial (field theory) — 0.90
• Higher local field — 0.90

Computed from structural-signature embeddings · 2026-09-08