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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.

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?

gjm11 2y ago
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.
- will-burner 2y ago
  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.
  - arjvik 2y ago
    For a layman (I stopped short of Galois theory so far), what’s different about the permutation groups of quintic roots and above that leads to this?
    - will-burner 2y ago
      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.

gjm11 2y ago

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.

will-burner 2y ago
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.
- arjvik 2y ago
  For a layman (I stopped short of Galois theory so far), what’s different about the permutation groups of quintic roots and above that leads to this?
  - will-burner 2y ago
    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.