The galois-theory tag has no usage guidance.

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### Irreducibililty tests for cubic and quartic polynomials over finite fields

The unpublished preprint:
D. G. A. Jackson, The irreducibility of a cubic over $\mathbb{F}_q$, Research Report 98-17, Univ. of Sydney (1998)
gives necessary and sufficient conditions (when ${\rm ...

**1**

vote

**1**answer

168 views

### Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...

**17**

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**1**answer

990 views

### Janelidze's Galois theory

I am interested in learning about categorical Galois theory, as developed by Janelidze. I am a graduate student who has good familiarity with category theory, but not in the level of doing research on ...

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180 views

### on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field.
$F=k((t))$ and $\mathcal{O}=k[[t]]$.
Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$.
Let $E=k((t^{1/n}))$ with $n$ prime to the ...

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**2**answers

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### Galois Group of $x^n-2$

Let $n \in \mathbb{N}$, then the order of the Galois Group
of $x^n-2$ coincide with $n \phi(n)$ for $n\in \{ 1 , \dots , 36 \}$
except for $n=\{ 8, 16, 24, 32 \}$ where this order is $\frac{ n \phi ...

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**2**answers

207 views

### Confusion about two statements about cohomology of curves with automorphisms

Let $\pi :C\rightarrow \mathbb{P}^{1}$ be a cyclic cover of degree $m$ of $\mathbb{P}^{1}$. So $C$ has an action of $\mathbb{Z}/m\mathbb{Z}$. Let $\xi$ be a primitive $m$-th root of unity. Consider ...

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**0**answers

54 views

### lift isomorphic in a sufficiently thick fiber

Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$.
We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$.
Let $d=val(\Delta(P))$ where ...

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**1**answer

430 views

### Relation between Galois theory and Etale Cohomology

I am a graduate student working on category theory. I am familiar with categorical Galois theory, in the way developed by Janelidze - as described for example in "Galois Theories". I am now trying to ...

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**1**answer

210 views

### Deciding if the largest absolute value real root lies in a cyclotomic extension

Given an algebraic equation of degree $n$ of form: $$x^{n} - a_{n-1}x^{n-1} - a_{n-2}x^{n-2} - \dots - a_{0} = 0$$ where each $a_{i} \in \Bbb Q^{+}$ and atleast one positive root, how does one decide ...

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348 views

### About Schanuel's conjecture

I just read an article of Ram Murty about transcendence of special values of L-functions, and it seems that Schanuel's conjecture plays a crucial role in it. So given a positive integer $n$, let's ...

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247 views

### Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success.
Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...

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**1**answer

620 views

### Embeddings of $\overline{\mathbf{Q}}$ into $\mathbf{C}$

Keenan Kidwell's answer to Place stabilizers for the absolute Galois Group mentions that "choosing a complex conjugation" in $G_{\mathbf{Q}}$ means choosing an embedding ...

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**1**answer

313 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 ...

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**1**answer

526 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 ...

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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, ...

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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 ...

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231 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 ...

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**1**answer

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 ...

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**0**answers

111 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 ...

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**1**answer

353 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. ...

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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 ...

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**1**answer

198 views

### a question of local field

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 obtain an action of ...

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**1**answer

664 views

### Galois group of constructible numbers

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 obtainable by ...

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46 views

### IGP for non-fixed ground field

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 given in a first ...

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611 views

### Embeddings of finite groups into GL(n,Q_p)

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 that there are ...

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517 views

### Is there a nice criterion for when the splitting fields of two irreducible polynomials are equal?

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 criterion for knowing ...

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1k views

### Books on advanced galois theory

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/1584883936. I am ...

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236 views

### Notation Problem, Fixed Rings and Fields

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 merit an answer.
In ...

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319 views

### Galois descent for semilinear endomorphisms

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 an $\sigma$-linear ...

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179 views

### Hurwitz's construction of simple covers

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

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85 views

### solvablity for some polynomial

We know that if F is a field which ch(F)=0,p(x) is a polynomial with coefficient of F,then p(x)root solvablity if and only if the Galois group of p(x) is solvablity .Here I want to know if the ...

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223 views

### Correspondences of schemes induced by a finite Galois extension

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 schemes.
Recall that if ...

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**1**answer

201 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 ...

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**1**answer

449 views

### How does an irreducible polynomial of prime power order split over an extension of prime power degree

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 was another ...

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805 views

### why are subextensions of Galois extensions also Galois?

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 ...

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398 views

### Primitive Elements for $S_n$ Galois Extensions?

This is an offshoot of my other question two days ago.
How to apply Hilbert's Irreducibilty theorem?
But it is of independent interest.
Solutions of Inverse Galois Problem for a finite group ...

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**1**answer

373 views

### How to apply Hilbert's Irreducibilty theorem?

I do not know how to correctly interpret Hilbert's Irreducibility theorem
with Galois group as my aim.
Here $K$ is a number field (or simply $\mathbf{Q}$).
Scenario 1: Take a field $L$ that is a ...

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votes

**1**answer

513 views

### Algorithm for determining whether two polynomials have the same splitting field

This question asks how to tell whether two cubic polynomials with coefficients in $\mathbb{Q}$ have the same splitting field. There are several answers to the question, but they don't include proofs. ...

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930 views

### Profinite groups as étale fundamental groups

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 connected scheme?
...

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670 views

### Is it known if the absolute Galois group is “divisible”?

The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...

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**1**answer

292 views

### Langlands Paper on representations of abelian algebraic groups

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his maps is obvious and ...

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1k views

### On a theorem of Galois

I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois :
Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...

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169 views

### power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m
can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...

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**1**answer

606 views

### When does a polynomial split over Q?

If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?!
In another words, what could you impose on the coefficients to ...

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435 views

### Does this isomorphism between Galois groups hold for transcendental extensions?

Suppose $L/K$ and $M/K$ are algebraic extensions of a field $K$, such that $L\cap M=K$, and $L/K$ is a normal extension. It is well-known that, with these conditions, we have:
$$\text{Gal}(L/K)\cong ...

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**1**answer

351 views

### is there any bound on the absolute number of algebraic integer in terms of its degree?

If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))?
Clearly we need to have distinct roots of unity otherwise this ...

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765 views

### Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...

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### Solving z^n=a+ib using only radicals of positive real numbers

Let $a+bi\in\mathbf{C}$ be a complex number with $a,b\in\mathbf{R}$. Then
it is easy to find exact solutions of $z^2=a+ib$. For example let $z=u+iv$. Then
$$
u^2+v^2=z\overline{z}=\sqrt{a^2+b^2}
$$
...

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418 views

### Place stabilizers for the absolute Galois Group

Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...

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304 views

### A cubic polynomial which contains a linear factor with irreducible residual quadratic form

Let $f(x)\in\mathbb{Z}[x_{1},\dots,x_{n}]$ be a cubic homogeneous polynomial, which factors as $f(x)=g(x)h(x)$ over $\mathbb{C}$ with $\mathrm{deg}(g)=1$ and $h$ irreducible over $\mathbb{C}$. Assume ...