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
In Branch
Source Primitive element theorem (Wikipedia)
Proved By
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.
In Group: Field Theory and Galois Theory, Lead section
In field theory, the primitive element theorem states that every
Proved By: Ernst Steinitz, Lead
The primitive element theorem was proved in its modern form by Ernst Steinitz, in an influential article on field theory in 1910, which also contains Steinitz's theorem; Steinitz called the "classical" result Theorem of the primitive elements and his modern version Theorem of the intermediate fields.
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