Glossary of Field Theory - Galois Theory

Galois Theory

Galois extension
A normal, separable field extension.
Galois group
The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.
Kummer theory
The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theory of quadratic extensions.
Artin-Schreier theory
Covers an exceptional case of Kummer theory, in characteristic p.
Normal basis
A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.
Tensor product of fields
A different foundational piece of algebra, including the compositum operation (join of fields).

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