Let $K$ be a local field with mix char, $k$ residue field. We have an exact sequence $0 \longrightarrow I \longrightarrow G_{K} \longrightarrow G_{k} \longrightarrow 0$ Then we o …

Let $\mathcal{C}$ be the field of constructible numbers, that is, the complex numbers constructible by compass and straightedge. It can be shown that it consists of all the numbers …

This question is inspired by some interesting comments on this recent question. Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}_n(\mathbf{C})$. It is known tha …

I am trying to make sense of the notation and certain sets in two articles by Annick Valibouze whose results I'm using for my bachelor's thesis, I hope it's relevant enough to meri …

I have an assignment to show the known result that any finite group occurs as Galois group of $k(x_1,...,x_n)/F$ for some field $F$. This seems like an insurmountable task to be gi …

How to get a Galois extension of the commutative semi-local ring $R=\mathbb{F}_2+v\mathbb{F}_2$, where $v^2=v$. The Galois extension of $R$ of degree $d$ is $\mathbb{F}_a+v\mathbb …

This question is a bit vague, but I was wondering if someone might have an insightful answer. Let $f_1$ and $f_2$ be irreducible polynomials in $\mathbb{Q}[x]$. Is there an easy c …

I have been studying galois theory on my own and find it very fascinating. I have gone through Ian Stewarts book: http://www.amazon.co.uk/Galois-Theory-Third-Chapman-Mathematics/dp …

Let $K \subset L$ be a finite Galois extension, $\sigma$ an automorphism of $L$ (not necessarily fixing $K$) and let $E$ be a finite-dimensional vector space over $L$ together with …

What is commonly meant by Hurwitz's construction of simple covers?

Does every profinite group arise as the étale fundamental group of a connected scheme? Equivalently, does every Galois category arise as the category of finite étale covers of a c …

Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\ …

I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between scheme …

Generally a Galois extension is defined to be an algebraic extension that is also normal & separable. It is then shown that in the sequence of field extensions $L|M|K$ if $L|K$ …

I asked this question in a similar form on math.se here, where it has been unanswered for a little over a week some work on it can be found there. The motivation for this question …