Fundamental theorem Galois Theory From time to time, I try to give my non-mathematician scientist friends descriptions of important theorems that I come across. 
One friend, watched a video on Galois and now I'm at a loss on how to state to her the Fundamental Theorem of Galois Theory in an intuitive yet mathematically correct manner. 
 A: A very powerful idea in mathematics is to reduce a problem in one setting to a problem in another setting. What at first might seem like a difficult problem in one setting can suddenly be revealed to be a relatively straightforward problem in another. These sorts of reductions pervade mathematics. 
The fundamental theorem of Galois theory performs a reduction of this type. The problem it reduces concerns the study of algebraic field extensions; roughly speaking, the idea here is to study nice collections of roots of polynomials, and the way in which the roots of some polynomials can be expressed in terms of the roots of other polynomials. If you've ever tried to work with polynomials that are not quadratic, you might have some appreciation for how difficult a problem this could have been.
Galois theory reduces this problem, in an appropriate sense, to the problem of studying the symmetries of the roots. In mathematics, we talk formally about symmetry using group theory. Galois theory associates to a polynomial the collection of all permutations of its roots which, roughly speaking, preserve all algebraic identities those roots satisfy. Roughly speaking again, this Galois group encodes in some appropriate sense the way in which roots of other polynomials relate to the roots of our polynomial, and because we know a lot about group theory it is often easier to study Galois groups than to study field extensions directly.
A remarkable result that can be proven using the fundamental theorem is the Abel-Ruffini theorem. Now, as it turns out, the quadratic formula that everyone is familiar with generalizes to a more complicated cubic formula and an even more complicated quartic formula. The Abel-Ruffini theorem states that these formulas do not generalize any further; there is no quintic formula expressing the roots of a quintic polynomial in terms of algebraic operations, including taking roots. 
The proof that proceeds via Galois theory reduces this problem to showing that the group of symmetries of $5$ things is more complicated, in a certain precise sense (it is not solvable), than the group of symmetries of $2, 3$, or $4$ things, and roughly speaking an analogue of the quadratic formula can only exist if the Galois group is not complicated in this particular way.
