Show that in the fundamental theorem of Galois theory, the subfield corresponding to the intersection of two subgroups and is the subfield generated by the corresponding intermediate subfields and .

Suppose is the subfield generated by and . Then here is intermediate subfield embarcing and .

For any such , and , so .

Thus .

Conversely, , . So .

The intersection of two intermediate fields and corresponds to the subgroup generated by the corresponding subgroups and .

The proof is very similar to the previous one.

Suppose we also know is Galois over the groud field and is another intermediate field. Then is isomorphic to .

By assumption, is normal. From group theory,

Theorem 2 says that the corresponding group to is . Since is normal then . And .

Theorem 1 says that the corresponding group to intermediate field is . And .

So the conclusion holds.

In fact this corollary is valid even is transcedental extension, namely

Let be a Galois extension of , let be an arbitrary extension and assume that are subfields of some other field. Then is Galois over and is Galois over . Let be the Galois group of over , and the Galois group of over . If then the restriction of to is in and the map

gives an isomorphism of on the Galois group of over .

Jacobson Algebra I. p243

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