The Galois Correspondence

Let L/K be finite Galois with group G. An intermediate field E determines the subgroup Gal(L/E), consisting of automorphisms fixing E pointwise. Conversely, a subgroup H determines its fixed field L to the H: the elements fixed by every member of H. Fixed elements form a field because automorphisms preserve sums, products, and inverses. The fundamental theorem states that these two operations are inverse bijections under the finite Galois hypotheses.

The phrase pointwise matters. An automorphism can carry a subfield onto itself while moving individual elements inside it. Such a map preserves the subfield as a set but does not belong to Gal(L/E). If E grows, fixing every element of E becomes a stronger restriction, so the subgroup shrinks. At the extremes, all of G fixes exactly K, while the identity subgroup fixes all of L. These endpoint checks quickly catch a reversed diagram.

For H corresponding to E, the degree from L down to E is the number of automorphisms in H. The degree of E over K is the index of H in G. Their product equals the degree of L over K, matching the tower law. A subgroup of order two in a group of order twelve therefore corresponds to a field of degree six over the base, not degree two. Say which end of the tower a degree measures.

Intersections and generated fields also reverse direction. The subgroup fixing the compositum of E and F is the intersection of their fixing subgroups: a map must fix both fields. The subgroup corresponding to E intersect F is generated by the two fixing subgroups. These statements let you compute an entire field lattice from a small subgroup lattice. They apply inside the fixed ambient finite Galois extension; changing the ambient field changes the group being used.

Every intermediate extension L/E is Galois, but E/K need not be. The additional condition is that H = Gal(L/E) be normal in G. Conjugating H transports its fixed field, so H is normal exactly when all base-field automorphisms preserve E as a set. In that case restriction gives a surjective homomorphism from G to Gal(E/K), with kernel H, and hence a quotient group G/H.

If H is not normal, the notation G/H denotes a set of cosets but does not define the quotient group claimed by the theorem. For instance, in an S3 splitting field of an irreducible cubic, the order-two subgroup fixing one root is not normal. Its fixed cubic field is consequently nonnormal over the base. Its degree is still the subgroup index, three; the correspondence continues to work even though the quotient-group conclusion does not.