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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 $G_{\...

**61**

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### Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:
Is it possible to express ...

**51**

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### The inverse Galois problem and the Monster

I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...

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### Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?

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### Forcing as a new chapter of Galois Theory

There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...

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### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

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### Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees $...

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### Which small finite simple groups are not yet known to be Galois groups over Q?

The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...

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### Is Galois theory necessary (in a basic graduate algebra course)?

By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...

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### Fields with trivial automorphism group

Is there a nice characterization of fields whose automorphism group is trivial? Here are the facts I know.
Every prime field has trivial automorphism group.
Suppose L is a separable finite extension ...

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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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### Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree.
Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$?
...

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### transcendental Galois theory

Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ of $K \setminus L$, ...

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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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### What are the possible sets of degrees of irreducible polynomials over a field?

Hopefully this is not too easy an exercise.
Let $F$ be a field. Let $I \subset \mathbb{N}$ be the set of all positive integers $d$ such that there exists an irreducible polynomial of degree $d$ over ...

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### Grothendieck says: points are not mere points, but carry Galois group actions

Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French).
The following is an excerpt from ...

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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 $p(...

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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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### Solving the cubic by “radicals” in characteristics 2 and 3

This question has no justification other than a bit of fun.
We all know that the cubic is solvable by "radicals" ($\root2\of{}$ and $\root3\of{}$) in characteristics $\neq2,3$. The formula was ...

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### Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between $...

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### 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 ...

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### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...

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### 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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### Profinite groups as étale fundamental groups

Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
...

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### 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 ...

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### Solubility of the quintic?

Over the p-adics, every Galois group is solvable. Does this imply that the quintic (and higher-order polynomials for that matter) can be solved by radicals over $\mathbb{Q}_p$?
EDIT: The original ...

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### Grothendieck's Galois theory without finiteness hypotheses

This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any ...

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### Number theory textbook based on the absolute Galois group?

I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting ...

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### Galois Group as a Sheaf

I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...

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### How to solve a quadratic equation in characteristic 2 ?

What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.
...

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### The Galois group of a random polynomial

Intuitively, the Galois group (of a splitting field over $\mathbb Q$ of) a polynomial $f\in\mathbb Q[X]$ taken at random is most probably the full permutation group on the roots of $f$. This intuition ...

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### 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 ...

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### Is the Leopoldt conjecture almost always true?

The famous Leopoldt conjecture asserts that for any number field $F$ and any prime $p$, the $p$-adic regulator of $F$ is nonzero. This is known to be equivalent to the vanishing of $H^2(G_{F'/F},\...

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### Is there an irreducible but solvable septic trinomial $x^7+ax^n+b = 0$?

The following irreducible trinomials are solvable:
$$x^5-5x^2-3 = 0$$
$$x^6+3x+3 = 0$$
$$x^8-5x-5=0$$
Their Galois groups are isomorphic to ${\rm D}_5$, ${\rm S}_3 \wr {\rm C}_2$ and
$({\rm S}_4 \...

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### How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...

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### Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as ...

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### 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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### Local Inverse Galois Problem

It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedian field $K$ has solvable (in fact supersolvable) Galois group $G$. One sees this by using the ramification filtration $...

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### Surprising Analogue of Q

I was describing Manish Kumar's work a few weeks ago to a fellow graduate student, and she stumped me with a big-picture question I couldn't answer.
Manish Kumar proved that the commutator subgroup ...

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### Is there a Galois correspondence for ring extensions?

Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...

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### On a theorem of Galois

I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois :
Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...

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### Janelidze's Galois theory

I am interested in learning about categorical Galois theory, as developed by Janelidze. I am a graduate student who has good familiarity with category theory, but not in the level of doing research on ...

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### Galois theory timeline

A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective ...

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### A Galois Theory Computation

Excuse me for the specificity of this question, but this is a silly computation that's been giving me trouble for some time.
I want to explicitly realize the order 21 Frobenius group over ℂ(x), ...

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### Galois theory for polynomials in several variables

I feel a bit ashamed to ask the following question here.
What is (actually, is there) Galois
theory for polynomials in
$n$-variables for $n\geq2$?
I am preparing a large audience talk on ...

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### Connes-Kreimer Hopf algebra and cosmic Galois group

Hi,
I'm interested in the relation between the two following constructions motivated by renormalization:
Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...

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### $2$-categorical structure in Grothendieck's Galois Theory

Grothendieck's Galois Theory, as developed in SGA I, V.4, or very gently in Lenstra's notes, establishes an equivalence between profinite groups and Galois categories. We can put this into the ...

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### On the solvable octic $x^8-x^7+29x^2+29=0$

The irreducible but solvable octic,
$$x^8-x^7+29x^2+29=0\tag{1}$$
was first mentioned by Igor Schein in this 1999 sci.math post. This does not factor over a quadratic or quartic extension, but over ...

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### On the inverse Galois problem

Q: What is the "simplest" finite group $G$ for which we don't know how to realise it as a Galois group over $\mathbf{Q}$ ?
So here the word simplest might be interpreted in a broad sense. If you ...

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### Is $x^{n}-x-1$ irreducible?

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.