Tagged Questions

4
votes
0answers
54 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
0
votes
0answers
32 views

Semilinear representations as modules?

Let $F$ be a field and let $G$ be a group. I often invoke module theory results to prove things about $F[G]$-modules, which then translate into results about linear $F$-representations of $G$. What ...
2
votes
2answers
77 views

Galois Extensions generated by Algebraic Representations

$\newcommand{\Q}{\mathbb Q}$ Let $F=\mathbb Q^{ab} \subset \mathbb C$, i.e. the algebraic numbers. Let $G$ be a finite group of order $n$ and let $\phi: G \rightarrow GL_m(F)$ be a representation. ...
1
vote
2answers
416 views

a visual route to learning Galois theory

I really like the ideas of Galois theory: that you can think about all the algebraic numbers you can make starting with some set of them that there is some structure to this set of "algebraically ...
4
votes
1answer
178 views

Local Langlands correspondence: Weil-Deligne group

While reading the book 'Langlands correspondence for loop groups', I came across the definition of the Weil group $W_F$ and the Weil-Deligne group $W'_F = W_F \ltimes \mathbb{C}$ with action ...
1
vote
1answer
244 views

What is a Weil-Deligne representation?

Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
1
vote
4answers
297 views

How to interpret the phrase “transforms under the irreducible representation”?

I'm reading Robert Gilmore's "Lie Groups, Physics, and Geometry," and trying to understand his brief presentation of Galois theory. I think I get the gist of the method, but would be grateful for help ...
7
votes
5answers
489 views

Shortest way of proving that the Galois conjugate of a character is still a character

Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ ...
7
votes
3answers
249 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...