3.6. Galois group of a polynomial.

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If the polynomial f ∈ F[X] is separable, then its splitting field E is Galois over F, and we

call Gal(E/F) the Galois group Gf of f.

Let f =

_n

i=1(X − αi) in the splitting field E. We know elements of Gal(E/F) map

roots of f to roots of f, i.e., they map the set {α1, α2, . . . , αn} into itself. Since they are

automorphisms, they define permutations of {α1, α2, . . . , αn}. As E = F[α1, ..., αn], an

element of Gal(E/F) is uniquely determined by its action on {α1, α2, . . . , αn}. Thus Gf can

FIELDS AND GALOIS THEORY 27

be identified with a subset of Sym({α1, α2, . . . , αn}) ≈ Sn. From the definitions, one sees

that Gf consists of the permutations σ of {α1, α2, . . . , αn} with the property

P ∈ F[X1, . . . ,Xn], P(α1, . . . , αn) = 0 =⇒ P(σα1, . . . , σαn) = 0.

This gives a description of Gf without mentioning fields or abstract groups (neither of which

were available to Galois).

Note that (Gf : 1) ≤ deg(f)!.

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