Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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A different proof of the insolubility of the quintic?

I'm familiar with the "standard" proof using Galois theory that there is no general formula for solving an equation of fifth (or higher) degree using radicals (i.e. arithmetic and root-taking). ...
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Galois theory with non-irreducible polynomials

I was browsing through Wikipedia for some reason or another, and it says that there is a way to do Galois theory without worrying about whether or not polynomials are irreducible. That's a bit ...
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Congruences mod primes in Galois extensions

I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields ...
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Powers of $x$ as members of Galois Field and their representation as remainders

first question on math.stackexchange :) I'm studying for a Cryptography - Communication Security exam, and it involves a certain quantity of number theory - finite field theory, so be warned: this is ...
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How to describe the Galois group of the compositum of all quadratic extensions of Q?

This is Problem 1.7 from Gouvea's lecture notes on deformations of Galois representations. In particular, he asks you to show that it has many subgroups of finite index which are not closed. So here's ...