The galois-theory tag has no usage guidance.

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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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.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

**0**answers

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**

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**0**answers

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**

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**2**answers

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

**2**answers

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**

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**2**answers

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

**1**answer

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

**0**answers

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 ...

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votes

**1**answer

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

**2**answers

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**

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**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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?
...

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**0**answers

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

**1**answer

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**

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**1**answer

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 ...

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**0**answers

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-...

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votes

**2**answers

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

**1**answer

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

**4**answers

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. ...

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votes

**1**answer

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\...

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**0**answers

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 ...

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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**

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**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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**

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**0**answers

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**

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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\...

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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 ...

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**0**answers

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 ...

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**1**answer

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

**1**answer

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 + ...

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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 $\...

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**1**answer

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 ...

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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 ...

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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

**1**answer

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 $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$

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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 $...

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**0**answers

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 $\sigma(x)=x$, ...