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51
votes
0answers
4k views

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
votes
0answers
673 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
votes
0answers
739 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
votes
0answers
275 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
votes
0answers
732 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
votes
0answers
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 ...
11
votes
0answers
305 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
votes
0answers
915 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 ...
11
votes
0answers
670 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 ...
9
votes
0answers
281 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 ...
7
votes
0answers
327 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
votes
0answers
734 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
votes
0answers
166 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 ...
6
votes
0answers
288 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
votes
0answers
209 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
votes
0answers
87 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 ...
5
votes
0answers
200 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 ...
4
votes
0answers
139 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
votes
0answers
174 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
votes
0answers
167 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
votes
0answers
291 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
votes
0answers
199 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 + ...
4
votes
0answers
527 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 ...
3
votes
0answers
172 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
votes
0answers
153 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
votes
0answers
187 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
votes
0answers
164 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
votes
0answers
330 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 ...
3
votes
0answers
386 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 ...
2
votes
0answers
154 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
votes
0answers
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
votes
0answers
90 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
votes
0answers
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
votes
0answers
332 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
votes
0answers
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
votes
0answers
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 ...
2
votes
0answers
728 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 ...
2
votes
0answers
514 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
votes
0answers
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 ...
1
vote
0answers
192 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
vote
0answers
340 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 ...
1
vote
0answers
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
vote
0answers
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
vote
0answers
178 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
vote
0answers
186 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
vote
0answers
300 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
votes
0answers
37 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
votes
0answers
101 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 ...
0
votes
0answers
171 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 ...
0
votes
0answers
222 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 ...