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Canonical static concept record

Normal field extension

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Summary

An algebraic extension $L/K$ in which every irreducible polynomial over $K$ having one root in $L$ splits completely in $L$ .

Record metadata

Canonical name: Normal field extension
Concept ID: normal_extension
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

algebraic extension $L/K$

Axioms / constraints

stability under $K$ -embeddings into an algebraic closure

Canonically induces

A splitting-field characterization

Concept sources

Wikipedia — Normal extension — Normal extension · encyclopedic reference · source ID wp-normal-extension
nLab — Normal field extension — Normal field extension · mathematical reference · source ID nlab-normal-extension
Milne — Fields and Galois Theory — Fields and Galois Theory · course text · source ID milne-fields-galois-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Algebraic field extension Normal field extension

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_algebraic_extension_to_normal_extension_impose_axiom
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: require normality

Authored explanation

Irreducible polynomials with one root in the extension must split there.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Milne — Fields and Galois Theory — Fields and Galois Theory · course text · source ID milne-fields-galois-theory

Galois extension Normal field extension

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_galois_extension_to_normal_extension_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is normal

Authored explanation

A Galois extension is normal and separable; forgetting separability leaves a normal extension.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Milne — Fields and Galois Theory — Fields and Galois Theory · course text · source ID milne-fields-galois-theory

Splitting field Normal field extension

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_splitting_field_to_normal_extension_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is a normal algebraic extension

Authored explanation

A splitting field of a family of polynomials over the base field is algebraic and normal over that field; forgetting the chosen polynomials leaves the normal extension.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Milne — Fields and Galois Theory — Fields and Galois Theory · course text · source ID milne-fields-galois-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.