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When I studied Galois theory, we exclusively used the correspondence between field extensions and automorphism groups to turn field theory problems into analogous group theory problems.

Are there any situations where it's useful to do this in reverse, i.e. using field theory to prove results about groups?

user3716267
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  • Related – Rushabh Mehta Jun 25 '21 at 18:56
  • Definitely works with topology or graph theory, e.g. using covering spaces (topology) or graph theory to prove the Nielsen-Schreier theorem. A purely group-theoretic proof exists, but is a bit fiddly. – Teddy38 Jun 25 '21 at 19:50
  • It is unclear what kind of applications you are interested in. For instance: Are you asking about using the field theory to establish some properties of Galois groups $Gal(K/F)$? Or, are you asking about applications to groups which, nominally, are not given as Galois groups of any field extension? – Moishe Kohan Jun 26 '21 at 17:04
  • @MoisheKohan I was imagining the former of your two suggestions, but either would be interesting. – user3716267 Jun 26 '21 at 17:43

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Ok, here is an example: The absolute Galois group $Gal(\bar{\mathbb Q})$ is not finitely generated.

Indeed: Consider the sequence of quadratic extensions of ${\mathbb Q}$ given by the polynomials $x^2-p=0$, where $p$'s are prime integers. By the Galois theory, each extension defines a homomorphism $\phi_p: Gal(\bar{\mathbb Q})\to {\mathbb Z}/2$. Then one verifies that these homomorphisms are linearly independent as elements of the vector space $$ Hom(Gal(\bar{\mathbb Q}), {\mathbb Z}/2). $$ From this, one concludes that the group $Gal(\bar{\mathbb Q})$ is not finitely generated.

Moishe Kohan
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  • Nifty! One thing I've been wondering about as I've learned more mathematics is the seeming wealth of tools for transferring theorems from one type of structure to another. It seems like there's a lot of "cleverness" involved in identifying which sorts of objects are well-equipped for a given proof. I'm hoping at some point I'll start to see more of a guiding structure there. – user3716267 Jun 26 '21 at 20:57