I was recently looking at algebraic proofs of FTA, and decided to review some Galois theory. So here’s some notes/an attempt to motivate Galois theory (perhaps up to or a bit before fundamental theorem of Galois theory), from the perspective of someone who much prefers field theory/polynomials to group theory/automorphisms (although maybe I should just stop being stubborn).
(Random side note: in the context of abstract algebra, polynomials are naturally motivated by “minimal (free) ring extensions”: if we have a commutative ring , then is the smallest ring containing with the additional element satisfying no relations. On the other hand, any constraints/extra relations would be polynomials, so at least for monic polynomials we get the quotient construction .)
Suppose is a singly-generated field extension (by primitive element theorem this is broader than it seems). If is the minimal polynomial of of degree , then let’s look at how it splits/factors in . If has some set of roots of roots lying in (the “splitting part”/linear factors of in ), say .
Generally, the main object of Galois theory seems to be splitting fields (or if we’re lazy, algebraic closures), but I’m still fighting through the material myself so I don’t fully appreciate/communicate this myself. Perhaps the point is just that it’s much easier to work concretely with roots (in algebraic closures) than directly with irreducible polynomials. (For example, we can then work with symmetric sums of the roots, and generally draw lots of intuition from number fields.)
We’ll work in characteristic for convenience, so e.g. the are pairwise distinct.
1. “Symmetry” of the : crux of Galois theory, and introducing Galois groups
The key is that (recall by definition), since the share minimal polynomials .
To make this symmetry precise, we phrase things in terms of -automorphisms of ; each -automorphism fixes coefficients of , hence is uniquely determined by sending . Thus they form a Galois group of size , since we easily check the automorphisms to be bijections of .
Continue reading Galois theory basics, part 1