Galois group

English

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

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

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

Usage notes

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

Translations

specific group associated with a field extension

• Arabic: زمرة غالوا
• Chinese:

  Mandarin: 伽罗瓦群

• French: groupe de Galois
• Hungarian: Galois-csoport (hu)
• Italian: gruppo di Galois m
• Japanese: ガロア群
• Korean: 갈루아 군 (gallua gun)
• Russian: группа Галуа (gruppa Galua)
• Spanish: grupo de Galois

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