Extensão de Galois

Em álgebra abstrata, uma extensão de corpo algébrica E/K se diz extensão de Galois (ou extensión galoisiana) se é uma extensão normal e separável. Neste caso, se pode considerar o grupo de Galois da extensão e sobre ele é válida a tese do Teorema Fundamental da Teoria de Galois.

Definição

Seja a extensão E sobre um corpo básico K (E/K).

  • Por ser normal, E é o corpo de decomposição de um polinômio com coeficientes em K; ou, equivalentemente, as K-imersões de E em um corpo algebricamente fechado que contenha K são automorfismos de E sobre K.
  • Por ser separável, este polinômio decompõe-se completamente em raízes simples.

Grupo de Galois

Ver artigo principal: Grupo de Galois

Sobre uma extensão de Galois E/K, se define o grupo de Galois Gal(E/K) como o grupo dos automorfismos de E sobre K. Por ser E/K normal, toda K-imersão entre E e Ω é um automorfismo e se tem:

sendo o cardinal do grupo .

Exemplos e contra-exemplos

  • A extensão não é de Galois; esta extensão não é normal.
  • Para qualquer número primo p, seja uma raiz primitiva p-ésima da unidade. Então a extensão é uma extensão de Galois. Esta extensão é o corpo de decomposição do polinômio p(x) = xp − 2.

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