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4
votes
1answer
501 views

Symplectic groups Sp_{2m}(2) as 2-transitive permutation (i.e. Galois) groups

Hello, I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms. Consider the block matrices ...
5
votes
0answers
758 views

“The Galois group of $\pi$ is $\mathbb{Z}$.”

Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question, that is, "The Galois group of $\pi$ is $\mathbb{Z}$.". In what ...
0
votes
1answer
205 views

The image of generator under an automorphism of a cyclic function field

I'm reading the proof of Lemma 4.1 [1] which says: "Let $F = K(x,y), y^q = f(x)$, where $q$ is a prime different from characteristic of $K$. Let $Z := Gal(F/K(x))$ and we have $Z < G < ...
0
votes
1answer
342 views

A question related to Hilbert's Irreducibility Theorem

My question is whether for every extension of number fields $L\subset K$, and for every $f_0(x),...,f_n(x)$ in $K[x]$, there is some $\alpha\in L$ such that ...
24
votes
2answers
1k views

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 ...
10
votes
3answers
949 views

Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?

Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$. Can we always find such an irreducible ...
1
vote
3answers
907 views

Constructive proof of algebraic elements forming a subfield

Let $F \leqslant E$ be a field extension. If $a, b \in E$ are algebraic over $F$ then $a+b$ and $ab$ are algebraic as well. There is an short proof of this using the extension $E(a,b)$: $[E(a,b):E]$ ...
15
votes
2answers
2k views

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 ...
1
vote
1answer
790 views

What does Gal(Q_p/Q) mean? [closed]

What does $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.) If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$, then does it have any property ...
6
votes
0answers
627 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 ...
12
votes
2answers
2k views

Why should the anabelian geometry conjectures be true?

I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation: If $X$ is a hyperbolic curve over some field ...
6
votes
1answer
324 views

Galois groups at closed points from Galois group at generic point?

Consider the finite map $\mathbb{A}^1_\mathbb{Q}\rightarrow \mathbb{A}^1_\mathbb{Q}$ given by $z\mapsto z^5-z$. The fiber over generic point is the field extension $\mathbb{Q}(t)[z]/(z^5-z-t)$ over ...
41
votes
1answer
2k views

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 ...
-4
votes
2answers
1k views

what part of using vieta's formulas violates quintic non-solvability? [closed]

You can write the n roots of an n degree polynomial in terms of its n coefficients, i.e., "Vieta's" formulas. You can solve this system of nonlinear equations using Newton's method and the Jacobian. ...
5
votes
5answers
1k views

Galois theory and algorithms

Steven Weintraub's book {\em A Guide to Advanced Linear Algebra} includes the following remark: "Of course, there is no algorithm for factoring polynomials, as we know from Galois theory." I can't ...
3
votes
1answer
482 views

What is the general statement of Hilbert 90?

I know two generalizations of Hilbert 90, but I don't if there is a statement that contains both: The first statement Let $K$ be a field, then $H^1(Gal(K), GL_n(K^{sep}))=0$. The second statement ...
6
votes
1answer
714 views

Quadratic extension of quadratic extension

Let $K_1$ be a perfect field. Let $K_2/K_1$ and $K_3/K_2$ be quadratic extensions. Let $K_4/K_3$ be the Galois closure of $K_3$ over $K_1$. Is it true that either $K_3 = K_4$ or $K_4/K_3$ is quadratic ...
0
votes
1answer
464 views

Multiplication of matrices in GF(2) and R

$H$ is an $n \times n$ matrix with elements in $ \{ -1,1 \}$ $G$ is an $n \times k$ matrix with elements in $GF(2)$ and also upper triangular, invertable $m$ is an $k \times 1$ vector with elements ...
20
votes
0answers
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 ...
2
votes
1answer
357 views

Searching for polynomials with squarefree discriminant

In connection with the Galois theoretic results surrounding the irreducibility of $f(x)= x^{N}-x-1$ over $\mathbb{Q}$, I've been trying to prove for a while that the discriminant of $f$ is actually ...
7
votes
1answer
1k views

Is there an alternative formula for solving cubic equations?

It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
14
votes
4answers
2k views

$Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$

Observe that we have $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$. More generally, assume that $K$ is a finite extension of Q. Is there any $\alpha \in K$ such that $K=Q(\alpha^n)$ for every $n \in N$?
4
votes
2answers
1k views

A family of polynomials with symmetric galois group

Consider the following family of polynomials in $K[x,y]$, where $K$ has characteristic zero: $f_n(x,y)=(x+y)^n+(x-1)y^n,$ for $n\geq 3$. I can prove that $f_n(x,y)$ has an irreducible factor of ...
5
votes
1answer
2k views

Motivation for the proof of Hilbert's Theorem 90

The proof of Hilbert's Theorem 90 about cyclic extensions goes like this: Let $\sigma$ be the generator of the Galois group of order $n$ and let $b$ have norm $1$, i.e. $b \sigma(b) \cdots ...
10
votes
2answers
872 views

What Dirichlet doesn't tell…

Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural ...
3
votes
1answer
783 views

Galois connections

I've been experiencing minor qualms about my preprint "A Galois Connection in the Social Network" (accepted by Mathematics Magazine, pending revisions), and one of them involves the way I describe the ...
7
votes
2answers
1k views

What are the different theories that the motivic fundamental group attempts to unify?

I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert. In ...
8
votes
3answers
715 views

What is the purpose of tangential base-points?

Let $V$ be an affine complex variety. Let $x \in V$ be a closed point. Then a tangential base-point at $x$ is $x$ together with a regular function $t$ on $V$ that is zero exactly on $x$ (to degree ...
5
votes
2answers
411 views

Expressing Galois actions on fundamental groups explicitly

Let $X$ be some variety over $\mathbb{Q}$, and let $\pi_1(X\times_{\mathbb{Q}}\mathbb{C},x)$ denote its (topological) fundamental group. As is well known $Gal(\mathbb{Q})$ acts on this fundamental ...
2
votes
1answer
155 views

In Riemann Existence, what is the interpretation of the space of complex-geometric points?

I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question: Question Riemann existence says that if we have a variety over $\mathbb{C}$, ...
22
votes
5answers
2k views

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 ...
16
votes
1answer
2k views

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 ...
6
votes
1answer
405 views

Splitting a polynomial with one root

Suppose we have an irreducible polynomial $f\in K[x]$. Is there some way to sometimes tell whether $f$ splits completely after adjoining just one root of $f$ to $K$? I am mostly interested in the ...
7
votes
1answer
717 views

Extensions obtained adding torsion points of an elliptic curve

When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, ...
5
votes
1answer
710 views

Effective Chebotarev density results for arbitrary number fields

So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of the ...
9
votes
2answers
541 views

On the field of invariants of a finite group

So let $G$ be a finite group and let $\iota:G\rightarrow S_n$ be an embedding of $G$ in a symmetric group of degree $n$ for some fixed integer $n$. Let $K$ be a fixed field of characteristic $0$. The ...
7
votes
1answer
892 views

What is a “best” transcendence basis for R/Q ?

It is easy to show, using the axiom of Zorn, that there exists a transcendence basis for $\mathbb{R}/\mathbb{Q}$, i.e. a set $S$, algebraically independent over $\mathbb{Q}$, such that ...
2
votes
1answer
283 views

nth-powers and degree n polynomials with coefficients in field extensions

Hi, Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$? Thanks
8
votes
3answers
2k views

Elementary Luroth theorem proof?

Hi, everyone! I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
16
votes
3answers
2k views

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 ...
11
votes
1answer
707 views

Is the etale fundamental group of Spec(Z)\{p_1,…,p_n} finitely presented?

(of course not, it's usually uncountable; I really mean is it the profinite completion of a finitely presented group). By definition, $\pi_1^{\operatorname{et}}(\operatorname{Spec}(\mathbb ...
3
votes
0answers
371 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 ...
47
votes
0answers
3k 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 ...
8
votes
2answers
1k views

Where is a good place to start learning about the Grothendieck-Teichmuller group?

I've had a desire to get some sort of handle on the Grothendeick-Teichmuller group for years now, but I've always felt that I could never find a source that was introductory and readable. The obvious ...
5
votes
1answer
376 views

Given a branched cover with branch cycle description $(g_1,…,g_r)$, does $g_i$ generate some decomposition group?

Classically: Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so ...
3
votes
1answer
289 views

Finite topological generation of Galois groups

Let $F/\mathbf{Q}$ be an extension of finite degree, and let $S$ be a finite set of places of $F$. Let $F_S/F$ be the maximal extension unramified outside $S$; what is the most natural way to see ...
13
votes
4answers
2k views

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 ...
12
votes
3answers
1k views

Finitely generated Galois groups

It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number ...
7
votes
2answers
576 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 $b \in F_r$ ...
12
votes
2answers
1k views

Using higher-order Bring radicals to solve arbitrary polynomials

It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...