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Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.

7 votes
Accepted

Which quartic fields contain the 4th roots of unity in their Galois closure?

As explained in the comments, the only non-trivial case is where $K/\mathbb{Q}$ has Galois closure $K'/\mathbb{Q}$ with $\operatorname{Gal}(K'/\mathbb{Q})$ isomorphic to $D_4$. I will do this case her …
R.P.'s user avatar
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12 votes
0 answers
265 views

Galois groups of classical differential equations

I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following: Is there a book or article devoted (either partially or completely) to …
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1 vote

Why do we make such big deal about the 'unsolvability' of the quintic?

I like this question because I agree with its sentiment. Let me give an additional reason why the insolubility of the quintic is an overrated result in my opinion. I believe that we shouldn't even be …
6 votes
Accepted

Reference request to proof that H$^2(\Gamma, \mathbb{Q}/\mathbb{Z}) = 0$

By the Galois cohomology long exact sequence, this is isomorphic to $\operatorname{H}^3(\Gamma,\mathbb{Z})$, and the vanishing of this is Chapter I, Corollary 4.17 in Milne's Arithmetic Duality Theore …
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7 votes
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Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?

The corestriction map on cohomology is indeed the norm in degree zero (see Tate's notes on Galois cohomology for example). By a dimension shifting argument, it then easily follows that the corestricti …
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1 vote

How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain...

The answer as to the surjectivity of $\alpha$ is no. As in algebraic number theory, the simplest way to prove that an element is not a norm is by local considerations. Let us consider $$ y=\frac{(x^3- …
1 vote

Irreducibility of polynomials over some number fields

Here is a different approach, which is arguably a bit more elementary. If $f=X^n-p$ splits in $K$, and $g$ is one of its factors, then the constant term of $g$, being a product of zeros of $f$, must b …
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