Fixed Fields, Galois Groups and the Galois Correspondence
Fixed Fields, Galois Groups and the Galois Correspondence: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Fixed Fields and Galois Groups
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
The Fundamental Theorem
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
Let G =Gal(E/F) be the Galois group of the extension E/F. If H is a subgroup of G, the fixed field of H is the set of elements fixed by every automorphism in H, that is, F(H) = {x ∈E : σ(x) = x for every σ ∈H}. If K is an intermediate field, that is, F ≤K ≤E, define G(K) = Gal(E/K) = {σ ∈G : σ(x) = x for every x ∈K}. I like the term “fixing group of K” for G(K), since G(K) is the group of automorphisms of E that leave K fixed.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
Let E/F be a finite Galois extension with Galois group G =Gal(E/F). Then (i) The fixed field of G is F; (ii) If H is a proper subgroup of G, then the fixed field of H properly contains F.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Proposition
Let E/F be a finite extension with Galois group G. If the fixed field of G is F , then E/F is Galois.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Theorem
Let f be a symmetric polynomial in the n variables X1, . . . , Xn. [This means that if σ is any permutation in Sn and we replace Xi by Xσ(i) for i = 1, . . . , n, then f is unchanged.] If e1, . . . , en are the elementary symmetric functions of the Xi, then f can be expressed as a polynomial in the ei.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Corollary
If g is a polynomial in F[X] and f(α1, . . . , αn) is any symmetric polynomial in the roots α1, . . ., αn of g, then f ∈F[X].
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Lemma
Dedekind’s Lemma The result that the size of the Galois group of a finite Galois extension is the degree of the extension can be proved via Dedekind’s lemma, which is of interest in its own right. Let G be a group and E a field. A character from G to E is a homomorphism from G to the multiplicative group E∗ of nonzero elements of E. In particular, an automorphism of E defines a character with G = E∗, as does a monomorphism of E into a field L.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Theorem
Fundamental Theorem of Galois Theory Let E/F be a finite Galois extension with Galois group G. If H is a subgroup of G, let F(H) be the fixed field of H, and if K is an intermediate field, let G(K) be Gal(E/K), the fixing group of K (see (6.1.1)). (1) F is a bijective map from subgroups to intermediate fields, with inverse G. Both maps are inclusionreversing, that is, if H1 ≤H2 then F(H1) ≥F(H2), and if K1 ≤K2, then G(K1) ≥G(K2).
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Theorem
Let E/F be a finite Galois extension and K/F an arbitrary extension. Assume that E and K are both contained in a common field, so that it is sensible to consider the composite EK. Then (1) EK/K is a finite Galois extension; (2) Gal(EK/K) is embedded in Gal(E/F), where the embedding is accomplished by restricting automorphisms in Gal(EK/K) to E; (3) The embedding is an isomorphism if and only if E ∩K = F.
Proof / verification strategy: Construct the natural group homomorphism, identify its kernel and image, and then use quotient structure to prove the claimed isomorphism or correspondence.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test subgroup, polynomial, root, field, homomorphism, automorphism, norm, Ext. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
- Assuming an algebraic extension is automatically normal or separable.
- Confusing the degree of a polynomial with the degree of an extension when the polynomial is not minimal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 6.1 | Fixed Fields and Galois Groups | 105–106 |
| 6.2 | The Fundamental Theorem | 107–109 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.