What is the overall idea of Galois theory?

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?

A triple series evaluating to $\sqrt{3}$

Is there a deeper meaning behind the "determinant" formula for the cross product?

Combinatorial interpretation of identity: $\sum\limits_{j=0}^b\binom{b}{j}^2\binom{n+j}{2b}=\binom{n}{b}^2$

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

Fourier Transform of a Polynomial

General relationship between braid groups and mapping class groups

Cardinality of power sets decides all of cardinal arithmetic?

What are the "numerator" and "denominator" of binomial coefficients called?

$A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated?

Is $e^e$ irrational?

Twin Primes (continued research)