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3
votes
1answer
765 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
933 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
667 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
403 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
153 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
395 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
684 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
674 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
524 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 ...
6
votes
1answer
869 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
280 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 ...
15
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
684 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
363 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 ...
45
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
986 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
372 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
286 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
565 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
994 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, ...
3
votes
1answer
395 views

What is the size of the smallest rigid extension field of the complex numbers?

Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case ...
1
vote
0answers
174 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
1answer
156 views

Behaviour of Primes under Regular Coefficient Extensions

Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
9
votes
2answers
966 views

Solvability in differential Galois theory

It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative. The proof I know goes as follows: Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
6
votes
2answers
522 views

Polynomial with Galois Group $D_{2n}$

How does one construct a polynomial with Galois Group $D_{2n}$? A general method would be preferable or if that's impractical then an example of it being done for any n would be appreciated. Thanks!
15
votes
1answer
900 views

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 ...
6
votes
3answers
613 views

Branch locus of the Galois closure of a Belyi morphism

A morphism of curves is said to be Galois if the corresponding extension of function fields is Galois. Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective connected curves over ...
6
votes
0answers
198 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 ...
1
vote
1answer
208 views

Image of spliting of short exact sequence of algebraic fundamental groups

If we have a variety X , over a field k, and x is a geometric point of X, and let $\bar x $be a geometric point of $X_{k^s} := X \times_k k^s $above x then we have the following short exact sequence: ...
3
votes
2answers
328 views

Rank of sum of Galois conjugates of a matrix

Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in ...
2
votes
1answer
363 views

Hurewicz theorem related to Galois group (or Tannakian categories)?

Is there a proof of the Hurewicz theorem $\pi_1(X)^{ab} = H_1(X, \mathbf Z)$ ($X$ a connected topological space) expressing $\pi_1(X)$ as the "Galois" group of $X$, i.e., group of deck transformations ...
4
votes
2answers
1k views

Image of norm map for local field

Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$. What is the image of the norm map $N_{E/F}$? In particular - ...
7
votes
3answers
1k views

Infinite simple Galois groups

Conjecturally, every finite group is the Galois group of some extension of the rationals. This question made me wonder what is known about infinite simple groups occurring as Galois groups. What ...
9
votes
2answers
422 views

Which polynomials arise as formulas for a conjugate

For any integer $r \geq 2$, et $V_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that $Q(\alpha)$ is a ...
7
votes
1answer
351 views

Is H^2(W_p,C^times) well-known?

Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated? (This group ...
7
votes
2answers
2k views

What are Galois Categories used for?

Galois categories are introduced (for the first time?) in SGA1, but here's an English introduction that's available online: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Lynn.pdf It ...
5
votes
2answers
369 views

What is an example of a regular realization of $C_5$ over $\mathbb{Q}(x)$?

It's known that all abelian groups are regularly realizable over $\mathbb{Q}(x)$, but it occurred to me that I don't even have an example of a cyclic regular extension of $\mathbb{Q}(x)$ handy. So: ...
13
votes
0answers
448 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 ...
2
votes
0answers
431 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 $L$ ...
5
votes
1answer
294 views

How many algebraic integers exist with degree $\leq k$ and bounds on the modulus of all Galois conjugates?

The precise question is the following: Question: Can one reasonably bound the number of algebraic integers $\alpha$ of degree at most $k$ - that means there exists a monic integer polynomial $p$ ...
4
votes
2answers
882 views

Cyclotomic extensions with split Galois group

$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$ Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that $$ \Gal(E/\Q) ...
2
votes
0answers
352 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 ...
3
votes
1answer
236 views

Are all solvable groups *regularly* realizable over Q(x)?

It is known for Hilbertian fields that all groups that are abelian, solvable, $A_n$ or $S_n$ are realizable over them. $\mathbb{Q}(x)$ is one such field, but it's not obvious that the extensions that ...
2
votes
3answers
2k views

What (permutation) groups can occur as galois groups of irreducible polynomials of degree n

I think the answers for the first few degrees ($n$) are: $n=2$, $S_2$ $n=3$, $S_3,A_3$ $n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group) $n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ...
18
votes
2answers
2k views

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 $c\neq 0$. ...