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 .
2. Looking at the splitting/linear part more closely, and introducing fixed fields
A priori, we know (by definition) that fixes (all elements of) . But if is small, then it’s reasonable that it might fix much more.
With or without this intuition, it’s natural to play around with these automorphisms. Of course, if for some , then for all (which are -automorphisms). So applying this to , we see that “permutes the factors of ”. Focusing on the linear factors, we have (recall the are distinct by characteristic ).
It follows that has coefficients fixed by , hence in the fixed field of (easily check it’s a field). So is a splitting field over (it’s Galois over ). Furthermore, since covers all the ( acts transitively on the ), fixes no proper subset of the ; in other words, by irreducibility, is the -minimal polynomial of .
3. Computational/explicit perspective on fixed field
Certainly we have . Does equality hold?
Well, certainly is irreducible in , hence certainly in as well. So because , we get , and by this forces .
Also, note that if two intermediate fields give the same min poly (of ), then this common set of -automorphisms is both and -automorphism. But you fix something in only if it’s in by a degree argument, so must have and vice versa. Alternatively, note that the poly lies in and is certainly irreducible so by degree argument we have .
4. Summary of results
Cool, so because the (splitting parts in of the min polys of ) are the same, the – and -automorphisms of are the same. In other words, . (Note that this is true for any field in between —the splitting parts are the same.)
But furthermore, is just large enough for for to lie in yet be irreducible. So finally,
As a corollary, since , we have and . I’d be interested in a direct derivation/interpretation of the . Also note that is a splitting field (of ) if and only if if and only if . I’ll probably have more to say later on, perhaps about intermediate fields.
(We can see some “competing goals” between these two paragraphs—in the former we want an intermediate field not too large, so the splitting part of the min poly stays the same; in the latter we want an intermediate field not too small, so the min poly completely splits. It’s quite nice that is just right.)
5. More on the thing
5.1. Good example to play with for the thing
A good example to play with is , . Then is simply between , so order , so . With the above approach, it’s not clear where , for instance, comes into play, where is just a third root.
5.2. Another perspective on
A different approach than the one above, that’s more satisfactory in a way: Take a primitive element with min poly , and again factor for of degree , with for a primitive element.
Extend to some (e.g. splitting field of over ) so that splits completely in , with . Let be the roots. Now partition these based on their . Suppose give , and give . Then consider the -isomorphism taking . It must map bijectively to , because corresponds to (and corresponds to ). So our partition splits guys into sets the same size as , and we get as desired.
5.3. A similar way to compare
A similar way to compare . Compare the factorizations of over and . Since the guys in are primitive elements (essentially by definition), maps to . In this proof it is natural to note that are the roots of lying in and , respectively, or equivalently that and for any respectively.