Galois group

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Galois group

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Etymology

Named after Évariste Galois, who first discovered them.

Noun

Galois group (plural Galois groups)

1. (algebra, Galois theory) The automorphism group of a Galois extension.
   • 1996, Patrick Morandi, Field and Galois Theory, Springer, page 123:

     In this section, we show how to determine the Galois group and the roots of an irreducible polynomial of degree 2, 3, or 4.

   • 2004, George Szeto, Liangyong Xue, “On Central Galois Algebras of a Galois Algebra”, in Alberto Facchini, Evan Houston, Luigi Salce, editors, Rings, Modules, Algebras, and Abelian Groups, CRC Press, page 493:

     Let ${\displaystyle B}$ be a Galois algebra over ${\displaystyle R}$ with Galois group ${\displaystyle G}$, ${\displaystyle C}$ the center of ${\displaystyle B}$, and ${\displaystyle K=\{g\in G\vert g(c)=c\ {\mathsf {for\ each}}\ c\in C\}.}$ A natural question is whether ${\displaystyle B}$ is a central Galois algebra with Galois group ${\displaystyle K}$.

   • 2009, Steven H. Weintraub, Galois Theory, 2nd edition, Springer, page 128:

     Our work actually gives an algorithm for computing Galois groups of polynomials ${\displaystyle f(X)\in \mathbb {Q} }$:

Usage notes

• The automorphism group of a Galois extension ${\displaystyle L/K}$ may be denoted ${\displaystyle {\mbox{Gal}}(L/K)}$.
• ${\displaystyle \operatorname {Gal} (F/F)}$ is the trivial group whose single element is the identity automorphism.

Translations

See also

• Galois algebra
• Galois closure
• Galois extension
• Galois theory

Further reading

• Galois theory on Wikipedia.Wikipedia
• Galois closure on Wikipedia.Wikipedia
• Galois extension on Wikipedia.Wikipedia