Galois group

Named after Évariste Galois, who first discovered them.

Galois group (plural Galois groups)

1. (algebra, field theory) A specific group associated with a field extension.
   1. Given a field extension E/F, the group Aut(E/F) of automorphisms of E that preserve elements of F.
   2. Given a field extension E/F, the group Aut(G/F), where G is the Galois closure of E.
   • 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 ${\displaystyle f(X)\in \mathbb {Q} [X]}$ :

• If ${\displaystyle E/F}$ is a Galois extension, the ${\displaystyle \operatorname {Aut} (E/F)}$ definition is used and the Galois group is usually denoted ${\displaystyle \operatorname {Gal} (E/F)}$ .
  • Otherwise, the ${\displaystyle \operatorname {Aut} (G/F)}$ definition is sometimes used.
• The group operation is function composition.
• ${\displaystyle \operatorname {Gal} (F/F)}$ is the trivial group whose single element is the identity automorphism.

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

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