This is the heart of the chapter (Section 14.2). It establishes a one-to-one correspondence between: of a Galois extension Subgroups of the Galois group
Another example: determining whether the roots of a polynomial generate a Galois extension. The solution would involve verifying the normality and separability. For instance, if the polynomial is irreducible and the splitting field is over Q, then it's Galois because Q has characteristic zero, so separable. Dummit And Foote Solutions Chapter 14
: Several mathematicians maintain partial or full solution manuals. Igor Van Loo's GitHub provides detailed steps for early sections of the chapter. Greg Kikola’s Guide This is the heart of the chapter (Section 14