proof of fundamental theorem of Galois theory

The theorem is a consequence of the following lemmas, roughly corresponding to the various assertions in the theorem. We assume L/F to be a finite-dimensional Galois extensionMathworldPlanetmath of fields with Galois groupMathworldPlanetmath


The first two lemmas establish the correspondence between subgroupsMathworldPlanetmathPlanetmath of G and extension fieldsMathworldPlanetmath of F contained in L.

Lemma 1.

Let K be an extension field of F contained in L. Then L is Galois over K, and Gal(L/K) is a subgroup of G.


Note that L/F is normal and separablePlanetmathPlanetmath because it is a Galois extension; it remains to prove that L/K is also normal and separable. Since L is normal and finite over F, it is the splitting fieldMathworldPlanetmath of a polynomialPlanetmathPlanetmath fF[X] over F. Now L is also the splitting field of f over K (because FKL), so L/K is normal.

To see that L/K is also separable, suppose αL, and let fFαF[X] be its minimal polynomialPlanetmathPlanetmath over F. Then the minimal polynomial fKα of α over K divides fFα, which has no double roots in its splitting field by the separability of L/F. Therefore fKα has no double roots in its splitting field for any αL, so L is separable over K.

The assertion that Gal(L/K) is a subgroup of G is clear from the fact that KF. ∎

Lemma 2.

The function ϕ from the set of extension fields of F contained in L to the set of subgroups of G defined by


is an inclusion-reversing bijection. The inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is given by


where LH is the fixed field of H in L.


The definition of ϕ makes sense because of Lemma 1. The


for all subgroups HG and all fields K with FKL follow from the properties of the Galois group. The fixed field of Gal(L/K) is precisely K; on the other hand, since LH is the fixed field of H in L, H is the Galois group of L/LH.

For extensionsPlanetmathPlanetmathPlanetmath K and K of F with FKKL, we have


so ϕ(K)ϕ(K). This shows that ϕ is inclusion-reversing. ∎

The following lemmas show that normal subextensions of L/F are Galois extensions and that their Galois groups are quotient groupsMathworldPlanetmath of G.

Lemma 3.

Let H be a subgroup of G. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    LH is normal over F.

  2. 2.

    σ(LH)=LH for all σG.

  3. 3.

    σHσ-1=H for all σG.

In particular, LH is normal over F if and only if H is a normal subgroupMathworldPlanetmath of G.


12: Since for all σG and αLH, σ(α) is a zero of the minimal polynomial of α over F, we have σ(α)LH by the of LH/F.

23: For all σG,τH the equality


holds for all xLH (from the assumptionPlanetmathPlanetmath it follows that σ-1(x)LH, which is fixed by τ). This implies that


for all σG,τH.

31: Let αLH, and let f be the minimal polynomial of α over F. Since L/F is normal, f splits into linear factors in L[X]. Suppose αL is another zero of f, and let σG be such that σ(α)=α (such a σ always exists). By assumption, for all τH we have τ:=στσ-1H, so that


This shows that α lies in LH as well, so f splits in LH[X]. We conclude that LH is normal over F. ∎

Lemma 4.

Let H be a normal subgroup of G. Then LH is a Galois extension of F, and the homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath

r:G Gal(LH/F)
σ σ|LH

induces a natural identification


By Lemma 3, LH is normal over F, and because a subextension of a separable extension is separable, LH/F is a Galois extension.

The map r is well-defined by the implicationMathworldPlanetmath 12 from Lemma 3. It is surjective since every automorphism of LH that fixes F can be extended to an automorphism of L (if LLH, for example, we can choose an αLLH such that L=LH(α) using the primitive element theorem, and we can extend σGal(LH/F) to L by putting σ(α)=α). The kernel of r is clearly equal to H, so the first isomorphism theoremPlanetmathPlanetmath gives the claimed identification. ∎

Title proof of fundamental theorem of Galois theory
Canonical name ProofOfFundamentalTheoremOfGaloisTheory
Date of creation 2013-03-22 14:26:38
Last modified on 2013-03-22 14:26:38
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 5
Author pbruin (1001)
Entry type Proof
Classification msc 12F10
Classification msc 11R32
Classification msc 11S20
Classification msc 13B05