# Tagged Questions

The tag has no usage guidance.

371 views

### Unramified extension and class field theory

I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me. Let $F$ be a ...
368 views

438 views

### Artin representations in Serre's book 'local fields'

Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group. In Serre's book 'local fields', chapter 6, a ...
234 views

### how do automorphisms act on the right in grothendieck's galois theory

So, I'm reading through some notes on the etale fundamental group (mostly Murre, but also some other notes I have), and I find it confusing how in a galois category $\mathcal{C}$ with fundamental ...
572 views

### Is there an algorithm to compute efficiently the dessin d'enfant from a Belyi pair?

Let $(X,f)$ be a Belyi pair, i.e. a Riemann surface $X$ together with a morphism $f: X \to \mathbb{P}^1$, ramified only in $0,1, \infty$. Grothendieck's dessin d'enfant is the pre-image $G$ of the ...
495 views

### Reduction modulo p of Galois groups

I'm studying the relationship between the Galois group of a polynomial with integer coefficients and the group of his reduction modulo $p$. More precisely, consider $\mathbb{K}$ a number field such ...
971 views

316 views

### Rigidity, moduli space, and moduli field

In his comment to the question Algebraic numbers and the complex projective line minus three points JSE says that for algebro-geometric objects defined over the complex numbers "in practice, 'the ...
536 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo $p$...
36 views

### Degree of factor of resolvent

As always with my questions this is not at research level, but the assertion is made in a research paper, plus no one's been able (or willing) to answer it over at MSE. Here is the original question, ...
1k views

### Frobenius density theorem

As mentioned by @MichaelZieve in his comment re Quadratic residue, Chebotarev's density theorem was preceded by an allegedly much easier theorem of Frobenius (Mike Zieve is certainly not the only one ...
233 views

### Finding the effective maximum number of subspaces in a finite dimensional vector space

Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search. For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...
326 views

### What evidence is there that $\mathbb{Q}^{ab}$ is ample?

A field $K$ is called ample if every smooth curve over $K$ that has a $K$-point has infinitely many $K$-points. Examples include fraction fields of henselian rings. (This is far from trivial, but was ...
113 views

### On Artin-Hironaka lemma and Galois theory

Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point. Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ...
382 views

### Gaussian Periods

I attempted to get some information on this from MSE, but did not even receive a comment, so i'm trying my luck here: Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. ...
1k views

### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials in the $A_5$ symmetries of the icosahedron (or dodecahedron)? Perhaps this is too vague a question. Q2. Are there ...
200 views

202 views

### Multiple eigenvalues over imperfect fields

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 $j_\alpha(A)$ for the ...
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 $L|K$ is Galois then $L|M$...