Galois group

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, 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 ${\\displaystyle f(X)\\in \\mathbb {Q} [X]}$:

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

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