The Primitive Element Theorem states that every finite separable field extension is simple, meaning the larger field can be generated over the smaller one by adjoining a single element, called a primitive element, rather than a set of several. In particular every finite extension of a field of characteristic zero, and every finite extension of a finite field, satisfies this condition automatically. It is a foundational structural result of field theory that simplifies the description of finite extensions used throughout Galois theory.
Facts
Partially Attested
Proof YearSteinitz proved the modern form in 1910; Galois sketched the splitting-field case earlier. Classification
Statement Form StatementEvery separable field extension of finite degree is simple. 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Primitive element theorem (Wikipedia)
Sources
1. Primitive element theorem (Wikipedia)
Introduction
Every separable field extension of finite degree is simple.
History
in an influential article on field theory in 1910
In Branch: Field Theory, Lead sentence
In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e.
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