Galois theory is the explanation and apex of theoretical math that you can motivate and talk about at a dinner table with people that don't even like math, lol.
Start with the quadratic formula, everyone seems to have some recollection of this. Talk about solving for x in polynomials. Then discuss if you can always solve for x, and what does that even mean. If you graph a polynomial it crosses the x-axis so there's a solution for x, but does that mean you can solve for it in a formula (this alludes to the fundamental theorem of algebra that every polynomial of degree n has n solutions in the complex numbers)?
It's tough to get the idea of solution by radicals and how that relates to what it means to have a formula for x in terms of the coefficients of the polynomial.
Anyways, the punchline is that there's no formula for x using basic arithmetic operations up to taking radicals, where the formula is in terms of the coefficients of the polynomial for a general degree 5 or higher polynomial. Galois theory proves this.
Galois is credited with this because it took a lot of imagination to think about how to formulate and prove that there is no formula. What does it mean to not have a formula? How do you formulate it properly and then prove it?
This isn't quite right -- Abel proved that there's no quintic formula before Galois came along. Galois theory gives a whole lot more insight, lets you understand why some quintics do have solutions in radicals, etc., but Galois doesn't (or at least shouldn't) get credited for proving that there isn't a quintic formula, because he wasn't the first to do that.
Don't let the truth get in the way of a good story! hahahaha
But yeah you're right
edit: i don't recall Abel's proof, but Galois reformulation of what it means to be solvable by radicals, introducing the permutation group of the roots is the big thing in my mind.
In lay terms the best I can say is that for n greater than or equal to 5, the set of all possible permutations of n things is complicated. For n less than 5 the set of all possible permutations of n things is simple just because n is small. That's what leads to there being general formulas for n = 2, 3, 4.
Galois translated whether a polynomial has a solution for x in terms of the coefficients using algebraic operations up to using radicals into a property of the group of permutations of the roots of the polynomial. The property of the group is whether the group is solvable. For n greater than or equal to 5, the general permutation group on n objects is not solvable but for n less than 5 is is. There just are not that many permutation groups for n = 2, 3, and 4 objects and all these permutation groups are solvable. Generically a group is not solvable and so we see this with larger n.
Start with the quadratic formula, everyone seems to have some recollection of this. Talk about solving for x in polynomials. Then discuss if you can always solve for x, and what does that even mean. If you graph a polynomial it crosses the x-axis so there's a solution for x, but does that mean you can solve for it in a formula (this alludes to the fundamental theorem of algebra that every polynomial of degree n has n solutions in the complex numbers)?
It's tough to get the idea of solution by radicals and how that relates to what it means to have a formula for x in terms of the coefficients of the polynomial.
Anyways, the punchline is that there's no formula for x using basic arithmetic operations up to taking radicals, where the formula is in terms of the coefficients of the polynomial for a general degree 5 or higher polynomial. Galois theory proves this.
Galois is credited with this because it took a lot of imagination to think about how to formulate and prove that there is no formula. What does it mean to not have a formula? How do you formulate it properly and then prove it?