What is the overall idea of Galois theory? I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening in Galois theory, and things seem to be a bit disconnected. I would be grateful if some could help me get an overall picture of the subject. The whole, I believe, is greater than the sum of its parts.
I am familiar with the concepts of field theory, such as extensions, separability, splitting fields, finite fields and all the other basic concepts you would expect to know to study Galois theory, and also have a fair understanding of the main concepts of Galois theory. I just dont get the whole picture. Thanks.
 A: Galois theory is a branch of abstract algebra that gives a connection between field theory and group theory, by reducing field theoretic problems to group theoretic problems. 
It started out by using permutation groups to give a description of how various roots of a polynomial equation are related, but nowadays, Galois theory has expanded to involve automorphisms of field extensions. 
It was motivated by looking for the roots of fifth degree polynomials in terms of the coefficients of the polynomial using algebraic operations and the application of radicals. Galois answered this question and gave us a method for examining/checking that an equation higher degree can be solved in this way. 
The epitome of Galois theory is the fundamental theorem of Galois theory. It describes the structure of field extensions. It says that for a finite and Galois field extension $E/F,$ there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.  
Does that help?
