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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
716 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
374 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 ...
50
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 ...
9
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
378 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
290 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 ...
14
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
3answers
605 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, ...
5
votes
1answer
481 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
180 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
159 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
1k 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
550 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!
17
votes
1answer
1k 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
639 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
203 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
217 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
333 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
377 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 ...
5
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
432 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
353 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 ...
8
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
381 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
470 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
477 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
305 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
902 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
357 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
244 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 $c\neq 0$. ...
5
votes
1answer
549 views

Is it possible to recover the degree of a field extension from a list of elements and the ground field?

I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
21
votes
4answers
1k views

$A_5$-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 ...
8
votes
1answer
488 views

When are the intermediate fields totally ordered?

The groups whose subgroups are totally ordered by inclusion are easy to classify; they are subgroups of $\mathbb{Z}/p^{\infty} = \text{colim } \mathbb{Z}/p^k$ for some prime $p$, and thus ...
9
votes
3answers
446 views

Congruences mod primes in Galois extensions

I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields ...
4
votes
5answers
1k views

Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?

Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...
2
votes
1answer
210 views

Linear independence in the algebraic closure of $\mathbb{C}(z)$

Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.) Define ...
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 ...
8
votes
2answers
1k views

When is sin(r \pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary. As the question suggests, I would like to know when ...
11
votes
1answer
1k views

History of the Normal Basis Theorem

The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space ...
11
votes
2answers
447 views

Realizing D_8 as a Galois group over C(x) with prescribed decomposition groups

Coming up with examples of $D_8$-covers of $\mathbb{C}(x)$ is easy. For example: $Quot(\mathbb{C}(x)[y,z]/(y^2=x(x-7), z^4=(y+\sqrt{-6})^2(y-\sqrt{-6})^2(y+\sqrt{-10})(-y+\sqrt{-10})^3))$ defines a ...
13
votes
2answers
876 views

Proof of the result that the Galois group of a specialization is a subgroup of the original group?

I have been using the following result: Given a polynomial $f(x,t)$ of degree $n$ in $\mathbb{Q}[x,t]$, if a rational specialization of $t$ results in a separable polynomial $g(x)$ of the same ...
31
votes
8answers
8k views

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) ...
3
votes
1answer
725 views

How do I visualize finite covers of curves over non-algebraically closed fields?

If $L$ is algebraically closed, fields of transcendence degree one over $L$ correspond to algebraic curves over $L$ up to birational equivalence, and finite extensions correspond to finite Galois ...