# Tagged Questions

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### Dummit and Foote on Galois and Representation Theory?

At some point, I'd like to learn both Galois Theory and Representation Theory. I currently know a lot of Group Theory and Linear Algebra, as well as some Ring Theory. I was thinking of reading ...
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### question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
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### Representation Theory versus Galois Theory. [closed]

At my university there is a debate about whether it is better to require students to have taken a class in Galois theory or to require a class in representation theory for admission into the graduate ...
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### Group ring of galois group [duplicate]

Suppose $E/F$ is Galois extension. What is it known about structure of $F[Gal(E/F)]$? I've learned only one fact in this direction - existence of normal basis in $E/F.$ But it's not truly about ...
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### Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
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### Homomorphisms from the additive groups of virtual characters into certain idele groups

This is a question from Frohlich's book 'Galois Module Structure of Algebraic Integers', Ch.1. Let $K$ be a number field and $\Omega_K=\text{Gal}(K^c/K)$ where $K^c$ is the separable closure of $K$. ...
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### Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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) ?
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### 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 ...
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 $\chi$ ...
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 ...