Galois groupとは 意味・読み方・使い方

名詞

Galois group (複数形 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, 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, Springer, 2nd Edition, page 128,

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

使用する際の注意点

• 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.