Introduction to Galois theory by Wilkins D.R.

By Wilkins D.R.

Similar abstract books

Intégration: Chapitres 7 et 8

Intégration, Chapitres 7 et 8Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce quantity du Livre d’Intégration, sixième Livre du traité, traite de l’intégration sur les groupes localement compacts et de ses functions.

Additional resources for Introduction to Galois theory

Example text

Now i does not belong to Q(ξ), since Q(ξ) ⊂ R. Therefore the polynomial x2 + 1 is the minimum polynomial of i over Q(ξ). 21 now shows that [L: Q(ξ)] = [Q(ξ, i): Q(ξ)] = 2. 18) that [L: Q] = [L: Q(ξ)][Q(ξ): Q] = 8. 46). Another application of the Tower Law now shows that [L: Q(i)] = 4, since [L: Q] = [L: Q(i)][Q(i): Q] and [Q(i): Q] = 2. 21). But ξ is a root of x4 −2. Therefore x4 −2 is irreducible over Q(i), and is the minimum polynomial of ξ over Q(i). 31 then ensures the existence of an automorphism σ of L that sends ξ ∈ L to iξ and fixes each element of Q(i).

50), and therefore [M : K] = |Γ(L: K)/H| = p. 57 that M = K(α) for some element α ∈ M satisfying αp ∈ K. 55). The induction hypothesis ensures that f is solvable by radicals when considered as a polynomial with coefficients in M , and therefore the roots of f lie in some extension field of M obtained by successively adjoining radicals. But M is obtained from K by adjoining the radical α. Therefore f is solvable by radicals, when considered as a polynomial with coefficients in K, as required. 58, we see that a polynomial with coefficients in a field K of characteristic zero is solvable by radicals if and only if its Galois group ΓK (f ) over K is a solvable group.

Now, given any root αi of f , there exists some σ ∈ G such that αi = σ(α). Thus if g ∈ K[x] is a polynomial with coefficients in K which satisfies g(α) = 0 then g(αi ) = σ(g(α)) = 0, since the coefficients of g are fixed by σ. But then f divides g. Thus f is the minimum polynomial of α over K, as required. Definition A field extension is said to be a Galois extension if it is finite, normal and separable. 45 Let L be a field, let G be a finite subgroup of the group of automorphisms of L, and let K be the fixed field of G.