The tag has no usage guidance.

learn more… | top users | synonyms

6
votes
2answers
410 views

Frobenius elements in infinite extensions

Let $K$ be a number field, $\bar K$ an algebraic closure and $G$ the associated absolute Galois group. How can I define the Frobenius elements of $G$ or at least their conjugacy class? I know how ...
3
votes
1answer
115 views

Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension. Now I'm interested in a more specific fact about Kummer extensions. From Hooley's paper "On Artin's conjecture", we ...
23
votes
2answers
583 views

Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction. Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
5
votes
2answers
325 views

Inverse Galois problem for simple Lie type groups

Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...
1
vote
1answer
116 views

Galois group - unknown word

I found in some articles a definition of the Galois group of a differential module. In the definition appeared the expression $(Repr(Gal(M,\nabla),Forget)$. According to the the articles, this pair ...
18
votes
1answer
1k views

Grothendieck's “La longue Marche à travers la théorie de Galois”

It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this. Is there any way to obtain a copy (online or not) of "La ...
5
votes
0answers
217 views

Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...
12
votes
0answers
304 views

Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
5
votes
2answers
547 views

Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?

I asked this question at MSE but I did not receive an answer. So I ask it at MO: We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by $Gal(f)$...
6
votes
2answers
405 views

Why are most coefficients of these minimal polynomials divisible by $p$?

For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ ...
8
votes
2answers
557 views

Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
6
votes
1answer
185 views

Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in \mathbb{R},...
6
votes
0answers
187 views

Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
-5
votes
1answer
302 views

Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...
5
votes
2answers
383 views

Finding the inertia group

Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$. What is the isomorphism class of the inertia group $I_p$, ...
5
votes
0answers
162 views

Expressing every algebraic number using roots of trinomials?

This question is a continuation of Is every polynomial a factor of a trinomial? We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...
8
votes
2answers
385 views

Inverse Galois problem for $GL_2$ of a compact local ring

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring) For which $A$ as above is there a ...
5
votes
0answers
205 views

On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups

Solving $a+b+c = abc = 6$ in the rationals entails solving, $$-24a+36a^2-12a^3+a^4=z^2\tag1$$ which is birationally equivalent to an elliptic curve. It can be shown that if $a$ is a solution, then ...
1
vote
1answer
91 views

Separable extensions of henselian fields

Let $(k,v)$ be a henselian field, with $\mathcal{O}$ and $\bar{k}$ being respectively its valuation ring and its residue field. If $K/k$ a finite separable field extension (on which $v$ thus extends ...
2
votes
2answers
417 views

Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

This is a re-post from MSE (because I did not get the kind of answer I wanted even after offering a bounty). At the outset I must mention that I don't have a fairly working knowledge of Galois Theory ...
5
votes
1answer
233 views

Subfields of $\mathbb{Q}\bigl(\sqrt[n]{a}\bigr)$ for $a>0$

This is related to a question on Math Stack Exchange. Given a rational number $a>0$ and an $n\in\mathbb{N}$ such that $x^n - a$ is irreducible over $\mathbb{Q}$, it is known that every subfield of ...
0
votes
1answer
186 views

Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation $a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 (...
2
votes
0answers
181 views

Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
6
votes
0answers
430 views

Algebraic Closure of the field of rational functions

Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is $\...
5
votes
1answer
159 views

Coverings/Cech cohomology of totally disconnected spaces

For any topological space $X$ we have a natural functor $\text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set})$ from the category of coverings of $X$ to the category of functors $\pi_1(X) \...
13
votes
1answer
561 views

When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
3
votes
0answers
191 views

How to handle a polynomial whose roots exhibit obvious symmetry

I've got a sequence of polynomials, and for each of them the roots obviously follow a definite pattern. Here are the roots of the 34th one All others have their roots arranged in a similar trident-...
32
votes
2answers
4k views

Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
5
votes
1answer
224 views

Are all rational exactly solvable differential equations known?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...
5
votes
4answers
816 views

Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false? We know a lot of things that would be true if the Riemann Hypothesis holds. ...
0
votes
1answer
228 views

Find all possible rational values of the parameter of a parametric cubic such that it is reducible

Description: Given the following parametric cubic polynomials ${E}^{3} - 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E + 135\, {\beta}_{\pm} \left({5\...
3
votes
0answers
181 views

Application of Stickelberger's Theorem to Quadratic field

I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...
14
votes
0answers
310 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
2
votes
1answer
185 views

What are some good references on the Galois theory, factorization, or minimality of differential equations?

Let $\lambda$ be a nonzero complex number and let $u(x)$ be some smooth function $\mathbb{R}\to\mathbb{C}$, not identically zero. I want to prove that if $u$ satisfies $$u'' + \lambda u'+ \lambda^2 u =...
3
votes
1answer
126 views

Density of tuples of conjugate algebraic numbers

One can see that algebraic numbers are dense in the complex plane by just looking at quadratic polynomials. I am interested in a "higher order" density of algebraic numbers. More specifically: is it ...
10
votes
1answer
239 views

Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$. Must $V$ have a solvable point? The variety $V$ is assumed ...
10
votes
2answers
471 views

Quintic polynomials generating cyclic extensions

We know that a cubic equation generates a cubic cyclic extension iff it has a perfect square discriminant. Now I am wondering if there is a similar condition for quintic polynomials. So I am trying to ...
25
votes
0answers
706 views

On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...
2
votes
0answers
179 views

Explicit formula for $\sin\frac{\alpha}{3}$ [closed]

Recently I was thinking on the following problem: find all $\alpha\in R$ such that $\sin\frac{\alpha}{3}$ has an algebraic expression involving only rational powers over rational constants and "$\sin\...
5
votes
0answers
176 views

On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$

I. Given the roots $x_i$ of the general cubic, $$x^3+c_2x^2+c_1x+c_0=0\tag1$$ with $c_i \in \mathbb Q$, it is easy to show that the expression, $$F_3 = (x_1^{1/3}+x_2^{1/3}+x_3^{1/3})^3$$ is an ...
11
votes
0answers
323 views

Evaluating products of cyclotomic polynomials at roots of unity

Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...
7
votes
1answer
343 views

abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of $$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$ obtained from the homotopy exact sequence of the étale ...
3
votes
1answer
121 views

Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by, $$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$ with discriminant $D = (7 + ...
0
votes
0answers
119 views

construct totally real cubic fields

Given some $e_i=0$ or $1$ for $1\le i \le 3$. I wish to construct a totally real cubic number field $K$ so that $\prod_{i=1}^3 sgn(\sigma_i(u))^{e_i}$ is always $1$ for any $u\in O_K^\times$ where $\...
9
votes
1answer
285 views

Variance of the roots of a complex polynomial

Let $P\in\mathbb{C}[X]$ be a complex polynomial of degree $n\geq 2$ with complex roots $\alpha_1, \alpha_2,\ldots, \alpha_n$. My question is about the existence of a formula for the variance of the ...
9
votes
0answers
356 views

Do we know that 'most' finite groups are Galois groups of number fields?

The inverse Galois problem is a classical problem in mathematics and asks whether every finite group can be realized as the Galois group of a finite field extension of the rational numbers. The ...
6
votes
2answers
587 views

Isomorphism problem for two radical extensions

Let $n\geq 2$ and let $a,b\in{\mathbb Q}$. Suppose that both the polynomials $A=X^n-a$ and $B=X^n-b$ are irreducible. We want to know whether ( * ) there is a root $\alpha$ of $A$ and a root $\beta$ ...
5
votes
1answer
463 views

Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
14
votes
2answers
1k views

Images of polynomials

Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all $...
2
votes
0answers
60 views

Finite extension of K(x) with extra structure: definable over field of invariants?

Let $K$ be an algebraically closed field, and let $\sigma$ be an automorphism on $K$. Set $k=K^\sigma$. Consider the rational function field $K(x)$ and extend $\sigma$ to $K(x)$ by $\sigma(x)=x$, ...