The galois-theory tag has no usage guidance.

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### Galois groups of enumerative problems

Consider Harris's 1979 paper, Galois Groups of Enumerative Problems (see here for the paper itself).
Given a problem in enumerative geometry, there is in addition to the obvious question of finding ...

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votes

**1**answer

206 views

### abelian and nonabelian parts of Aut($\widehat{F_2}$)

Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...

**3**

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**1**answer

164 views

### Polynomials of even degree with solvable Galois group

Let $f(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_0$ be a polynomial with integer coefficients and irreducible over $\mathbb{Q}$. For $n \geq 3$, $f$ is generically unsolvable by radicals. ...

**3**

votes

**1**answer

271 views

### Unramified extension of number fields

Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact?
Suppose we have an number field $K$, is any Galois ...

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### $A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...

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**2**answers

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### non-continuous inverse Galois problem

Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$.
Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that
$G/K$ is free of ...

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It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.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

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vote

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134 views

### Generic polynomial for alternating group ${A}_{4}$ is not correct

I was validating the percentage of cases where the generic two parameter polynomial for Galois group ${A}_{4}$ is valid. We have
\begin{equation*}
{f}^{{A}_{4}} \left({x, \alpha, \beta}\right) = ...

**6**

votes

**2**answers

349 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 ...

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**3**answers

1k views

### 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
$$
u^2+v^2=z\overline{z}=\sqrt{a^2+b^2}
$$
...

**3**

votes

**1**answer

105 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**

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**2**answers

493 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

301 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 ...

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vote

**1**answer

107 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 ...

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**2**answers

542 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 ...

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**1**answer

486 views

### Elementary proof for Hilbert's irreducibility theorem

I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding.
I am only interested in the simple case where the ...

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214 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 ...

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379 views

### Families of quintics in $\mathbb{Q}[x]$ with Galois group $A_5$

Theorem. The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root.
Question. What ...

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282 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 ...

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796 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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391 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 $\xi$ ...

**6**

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**1**answer

178 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 ...

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### 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 ...

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177 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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**1**answer

216 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 $k$ ...

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333 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 $I_p$, ...

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375 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 ...

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154 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 ...

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### 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 ...

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votes

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### Computing the Galois group of a polynomial

Does there exist an algorithm which computes the Galois group of a polynomial
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of ...

**13**

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497 views

### What are Mean Values of Ideal Densities in Galois Extensions?

In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote:
"Classifying extensions according to the Galois group
of their normal closure ...

**3**

votes

**1**answer

167 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 ...

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vote

**1**answer

62 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**

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379 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 ...

**0**

votes

**1**answer

181 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 ...

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### 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. ...

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### Automorphisms of a differential field and transcendence degree

Let $(\mathcal{F},+,\times,\partial)$ be a differential field, and let's define its automorphism group $Aut(\mathcal{F})$ as the group, under composition, consisting of all bijective maps ...

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164 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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263 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 ...

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**1**answer

501 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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**1**answer

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### 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) ...

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### 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 ...

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**1**answer

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### 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) ...

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### 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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**1**answer

223 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} ...

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**1**answer

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### Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...

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### 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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628 views

### Centralizers of elements in free profinite groups

I believe, although I can't say that I've given a rigorous proof, that for a free group $F_r$, and an element of it $a$, $C_{F_r}(\langle a \rangle)=$ the group generated by the elements $b \in F_r$ ...

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1k views

### more on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...

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### 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 ...