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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 ...

**24**

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**0**answers

699 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 ...

**18**

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765 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 ...

**15**

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307 views

### 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 ...

**14**

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309 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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755 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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504 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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297 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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705 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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322 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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950 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 ...

**10**

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188 views

### Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...

**9**

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181 views

### Weyl Groups as Galois groups

I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of ...

**9**

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347 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 ...

**8**

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792 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 ...

**7**

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371 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 $8$ ...

**6**

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158 views

### Characterizing regular Galois extensions by the set of their specializations

Let $E$ and $F$ be two regular Galois extensions of $\mathbb{Q}(t)$ with group $G$, and let $E_{t_0}$ resp. $F_{t_0}$ be the residue fields corresponding to specializing $t\mapsto t_0\in \mathbb{Q}$. ...

**6**

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184 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**

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381 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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304 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 ...

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216 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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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 ...

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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 ...

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204 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 ...

**5**

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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 ...

**5**

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222 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 ...

**4**

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165 views

### The density of quartic polynomials whose Galois group is a subgroup of $D_4$

Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois ...

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322 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 ...

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217 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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144 views

### Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting:
the minimal polynomial of the field over $\mathbb{Q}$, and
a decimal ...

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156 views

### splitting property of etale covering

Theorem (Global Splitting): Let $X$ be an integral separated normal scheme flat and of finite type over $\mathbb Z$. Let $\phi: Y\rightarrow X$ be a connected etale covering which splits completely ...

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187 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 ...

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173 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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564 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**

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214 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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398 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**

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407 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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172 views

### Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers.
Question 1: Is it true that $\mathbf{C}$ is ...

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176 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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59 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 $\sigma(x)=x$, ...

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100 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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183 views

### Parametric Solvable Septics?

Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 + 21x^5 + 35x^3 + 7x + a(7x^6 + 35x^4 + 21x^2 + 1)=0\tag{2}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - ...

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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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347 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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111 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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771 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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547 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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362 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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73 views

### Galois extensions inside a division ring

Let $D$ be a division ring which has finite dimension over its centre.
Q1. Under which conditions can one find a maximal subfield $K$ of $D$ and a proper subfield $L$ of $K$ such that $K/L$ is ...