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### Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmuller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here ...

**22**

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**0**answers

679 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 ...

**17**

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746 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 ...

**14**

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283 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?

**14**

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734 views

### Is there an infinite field of characteristic 2 whose multiplicative group is torsion free and (direct-sum) indecomposable?

Let $F$ be a infinite field of characteristic 2 whose multiplicative group $F^*$ is torsion free. I would like to conclude that $F^*$ is decomposable or find an example where $F^*$ is indecomposable.
...

**13**

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486 views

### Noether-Deuring for injections and surjections?

Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over ...

**12**

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262 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 ...

**12**

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678 views

### Does there exist a number field, unramified over a predetermined finite set of primes of Q, such that the inverse regular Galois problem is correct for that number field?

The question is: for any finite group, $G$, and any finite set of primes (of $\mathbb{Z}$), $P$, is there a number field $K$, such that there is a regular $G$-Galois extension of $\mathbb{P}^1_K$, and ...

**11**

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307 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 ...

**11**

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924 views

### Galois theory timeline (II)

This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...

**9**

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289 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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174 views

### Galois group for flexes of a plane $d - ic$

Here is the first problem considered in Harris's 1979 paper, Galois Groups of Enumerative Problems (see here for the paper itself).
The Galois groups are: for the flexes of a plane $d$-ic, the ...

**7**

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**0**answers

335 views

### The construction of the 257gon

If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...

**7**

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741 views

### Automorphisms of local fields

It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...

**6**

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171 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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227 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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291 views

### Cutting and pasting in Galois theory

I want to ask who was the first to use cut-paste construction in Galois theory.
This question is motivated from the trend in contemporary Galois theory to use patching methods to construct Galois ...

**6**

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210 views

### Involution on sextic polynomials?

The strangeness of $Aut(S_{6})$ suggests the following question:
Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be the zeros of a 6th degree polynomial $P$ over a field $F$.
Let ...

**5**

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208 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 ...

**5**

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184 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 ...

**4**

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150 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 ...

**4**

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170 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 ...

**4**

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296 views

### questions about the “relative fundamental group” of SGA 1 Expose XIII

$\newcommand{\LL}{\mathbb{L}}$
I'm reading SGA 1, obtained from http://arxiv.org/abs/math/0206203
My questions regard 4.5 and 4.5.1 (page 309) of Expose XIII.
Following "Exemples 4.4" in Expose ...

**4**

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201 views

### Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?

Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$?
For example, we have the well-known modular equation,
$$u^6 - v^6 + ...

**3**

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175 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 ...

**3**

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**0**answers

158 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 ...

**3**

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538 views

### Differential Galois number theory

Following http://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...

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188 views

### On the peculiar Lagrange resolvent of the septic $7x^7+14x^4+7x^3-1=0$

Given an irreducible solvable equation $P(x)=0$ of prime degree $p>2$ with rational coefficients and $\zeta^p=1$, define the usual Lagrange resolvents of the roots $x_i$ as,
$$R_n = ...

**3**

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168 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 ...

**3**

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336 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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388 views

### Norms in Galois extensions

Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$.
Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$,
where both ...

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156 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?
...

**2**

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

**2**

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93 views

### Galois correspondence for action of general linear group on purely transcendental extension

For a fixed positive integer $n$, the group $G=GL_n(\mathbb{C})$ acts on the field $K=\mathbb{C}(t_1,\ldots,t_n)$ by linear change of variables. I would like to know if there is something like a ...

**2**

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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 ...

**2**

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336 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 ...

**2**

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107 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 ...

**2**

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94 views

### Explicit defining equations for the Leopoldt locus

Let $F$ be a number field, which we assume for simplicity to be Galois and totally real. Set $\mathcal{O}_p=\mathcal{O}_F\otimes_{\mathbf{Z}}\mathbf{Z}_p$. The norm map on $\mathcal{O}_F$ extends ...

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733 views

### Cubic polynomials with “nice” roots, which can be expressed by trig functions of rational angles

Consider the cubic polynomial $x^3-ax+b$ for $a,b\in\mathbb N$.
It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ...

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517 views

### Decomposition group vs Galois group of completed extension for height > 1 primes

Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...

**2**

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358 views

### What do you call an algebraic element with the property that the generated field extension is normal?

Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you ...

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207 views

### Discriminant of a compositum of number fields, a bound?

Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension ...

**1**

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346 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 ...

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125 views

### Parametric Solvable Septics?

Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - 1)x^2(x + 1)^2=0\tag{2}$$
$$x^7 + 7x^6 - 28(n^2 + 27)x^2 + 112(n^2 + ...

**1**

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34 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, ...

**1**

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179 views

### $f(x_1,x_2,x_3,\ldots,x_n)$ Maximum how many different results can have with all permutation of inputs?

$\alpha _n=e^{2 \pi i/n}$
$$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$
Maximum how many different results can have with all permutation of ...

**1**

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188 views

### Decomposing anticyclotomic characters

Suppose $K/\mathbf{Q}$ is an imaginary quadratic field and $\chi$ is a finite-order character of $G_K=\mathrm{Gal}(\overline{K}/K)$ which is anticyclotomic, i.e. $\chi^{\sigma}:=\chi(\sigma g ...

**1**

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303 views

### Splitting of prime ideals in non-Dedekind domains?

This is a follow-up to this question. So that you don't have to flick back and forwards I'll briefly summarize:
My original question was on how to prove that a polynomial $g(x)$ obtained from ...

**0**

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39 views

### 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 ...

**0**

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105 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 ...