All Questions
31 questions
3
votes
0
answers
89
views
Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
- 605
1
vote
0
answers
86
views
Is the Galois closure of a $p$-adic Lie group extension also a $p$-adic Lie group extension?
Let $p$ be a prime. Let $K$ be a number field and $L/K$ be an infinite extension which is not necessarily Galois. Suppose that the automorphism group $\text{Aut}(L/K)$ of $L/K$ is $p$-adic analytic, i....
- 667
4
votes
1
answer
246
views
How do "Kummer closures" of fields look?
Let $F$ be a field and $A$ a finite abelian group. You can ask: does the regular representation $F[A]$ of $A$ split as a direct sum of 1-dimensional representations? This is equivalent to the ...
- 54.6k
2
votes
3
answers
345
views
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$
of cardinality $2m\ge 6$ where $m$ is odd.
Question 1. Is it true that $G$ always has a subgroup $H$ of index 2
...
- 14.1k
5
votes
1
answer
513
views
Learning Inverse Galois Theory
Can someone give me a roadmap for learning Inverse Galois theory?
I am a PhD student in the representation theory of finite groups. I studied Galois theory when I was an undergraduate student. The ...
- 195
4
votes
1
answer
243
views
Existence of intermediate field extensions for tamely ramified p-adic extensions
Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
- 421
3
votes
1
answer
384
views
Using the Lehmer quintic to solve $11$-degree equations and higher?
(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
- 12.6k
7
votes
2
answers
439
views
A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics
I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
- 12.6k
3
votes
1
answer
250
views
On the refined minimal ramification problem for $p$-groups
Let $p$ be a prime. The minimal ramification problem is to ask whether or not every finite $p$-group $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly ...
- 667
2
votes
2
answers
381
views
On V. Arnold's trinities regarding PSL(2,5), PSL(2,7), and PSL(2,11)?
Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}.$ Then the following functions for levels $...
- 12.6k
0
votes
0
answers
80
views
Projection map $\pi:\left(\mathcal{O}/n\mathcal{O}\right)^\times \to\left(\mathcal{O}/\gcd(n,m)\mathcal{O}\right)^{\times}$ of a CM elliptic curve
In this paper the author has mentioned in page $693$ under section $2.2$ that for an Elliptic curve $E/\mathbb{Q}$ with CM by an order $\mathcal{O}$ of an imaginary quadratic field $K$ there is a ...
- 309
6
votes
0
answers
375
views
How to construct this non-geometric mod $p$ Galois representation?
Let $ K $ be a totally real cubic field. I am concerned with the following kind of Inverse Galois Problem:
Is there a prime $ p>5 $ and a continuous representation $ \bar{\rho}:G_{K}\to {\rm
GL}_{...
- 863
-2
votes
1
answer
504
views
In Galois theory, why solvable groups must have their quotient groups be Abelian? [closed]
The definition of solvable groups can be regarded as two constraints, one is that there must be a sequence of normal subgroups, and the other is that the quotient groups between these sequences are ...
- 1
13
votes
1
answer
460
views
Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?
Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\...
user130124
5
votes
1
answer
1k
views
On progress towards inverse Galois problem over rationals
I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$.
From where I can read ...
- 591
1
vote
2
answers
1k
views
Is the absolute Galois group the same as the automorphism group? [closed]
Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\...
- 1,613
13
votes
4
answers
2k
views
Which groups are Galois over some p-adic field?
Suppose I have some finite $p$-group $G$, or a little extension of it.
How do I know if there exists a prime $l$ and a finite extension $K$ of $\mathbb{Q}_l$ such that $G$ is the Galois group of ...
- 11.3k
3
votes
1
answer
410
views
What do we know about these subgroups of $S_n$?
For each positive integer $n$, write $S_n$ for the symmetric on $n$-letters. Suppose that $m | n$ is a proper divisor of $n$, and write $n = km$. Consider the element
$$\displaystyle u(m,n) = \...
- 26.9k
6
votes
2
answers
622
views
Inverse Galois problem for simple Lie type groups
Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...
- 17.6k
10
votes
2
answers
723
views
Does the Galois group of a Pisot polynomial contain the alternating group?
Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...
- 11.3k
7
votes
4
answers
1k
views
Consequences of the Inverse Galois Problem
Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...
7
votes
2
answers
1k
views
Galois groups and prescribed ramification
What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
- 11.3k
14
votes
0
answers
1k
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.
...
- 575
20
votes
1
answer
2k
views
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 ...
- 563
5
votes
2
answers
683
views
Reducing 12th degree eqns (12T179) to an 11th degree eqn
I always wondered if the fact that the quartic can be solved by a cubic can be generalized to other even degrees $n$, namely if there is an ordering of the roots $x_i$ of form $x_1x_2+x_3x_4+\dots+x_{...
- 12.6k
12
votes
2
answers
893
views
Embeddings of finite groups into GL(n,Q_p)
This question is inspired by some interesting comments on this recent question.
Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}_n(\mathbf{C})$. It is known that there are ...
- 20.8k
20
votes
0
answers
1k
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 ...
- 19.2k
2
votes
2
answers
1k
views
Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
6
votes
5
answers
3k
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 ...
- 71
29
votes
3
answers
4k
views
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 ...
- 12.6k
73
votes
2
answers
8k
views
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 $M_{23}$, ...
- 4,994