Applications
The theorem converts the difficult-sounding problem of classifying the intermediate fields of E /F into the more tractable problem of listing the subgroups of a certain finite group.
For example, to prove that the general quintic equation is not solvable by radicals (see Abel–Ruffini theorem), one first restates the problem in terms of radical extensions (extensions of the form F(α) where α is an n-th root of some element of F), and then uses the fundamental theorem to convert this statement into a problem about groups that can then be attacked directly.
Theories such as Kummer theory and class field theory are predicated on the fundamental theorem.
Read more about this topic: Fundamental Theorem Of Galois Theory