Galois extension

English

Etymology

Named for its connection with Galois theory and after French mathematician Template:W.

Noun

1. Template:Lb An algebraic extension that is both a normal and a separable extension; equivalently, an algebraic extension E/F such that the fixed field of its automorphism group (Galois group) Aut(E/F) is the base field F.

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Usage notes

• Given an algebraic extension $E/F$ of finite degree, the following conditions are equivalent:
  • $E/F$ is both a normal extension and a separable extension.
  • $E$ is a splitting field of some separable polynomial with coefficients in $F$.
  • $|\mathrm{Aut}(E/F)|=[E:F]$; that is, the number of automorphisms equals the degree of the extension.
  • Every irreducible polynomial in $F[x]$ with at least one root in $E$ splits over $E$ and is a separable polynomial.
  • The fixed field of $\mathrm{Aut}(E/F)$ is exactly (instead of merely containing) $F$.

Hypernyms

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Derived terms

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Related terms

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Translations

• French: Template:T, Template:T
• Italian: Template:T

Further reading

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• Galois extension on Template:W