All Questions
Tagged with galois-theory gr.group-theory
73 questions
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
42
votes
2
answers
4k
views
Abel and Galois (and Arnold)
Question Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
user6976
41
votes
3
answers
3k
views
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 ...
- 151k
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
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
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
19
votes
1
answer
3k
views
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 ...
- 20.8k
17
votes
3
answers
2k
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 theory....
- 1,418
17
votes
1
answer
448
views
The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$
Given two positive integers $n,m$, which positive integers $d$ appear as the degree of $\mathbb{Q}(a,b)$ for two algebraic numbers $a$ and $b$ of degrees $n$ resp. $m$?
Two necessary conditions are $\...
- 5,482
15
votes
1
answer
761
views
Permutation Groups Containing non-commuting $p$-cycles
I noticed that the following is true, and that there is a reasonably elementary proof of it (in particular, the classification of finite simple groups is not needed). Let $G$ be a finite permutation ...
- 44.4k
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
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
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
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
11
votes
2
answers
2k
views
How to show the galois group of a polynomial is not an alternating group?
I am trying to prove that a certain class of polynomials have symmetric galois group.
Using the Newton polygon, I have shown that the galois groups of these polynomials are transitive on $k$-sets for ...
- 1,327
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
10
votes
2
answers
932
views
On the Galois group of the compositions of polynomials
We reprint an old math SE question here (see https://math.stackexchange.com/questions/1241224/composition-of-polynomials-and-galois-theory):
"
Let $f(x)$ be a polynomial of degree $n$ over $\...
- 1,755
9
votes
1
answer
410
views
Galois embedding question for dihedral groups
Let $D_n$ be the dihedral group of order $2n$. Then all the quotients of $D_n$ are dihedral as well, and of the form $D_k$ with $k \mid n$. So for a field $K/\mathbb{Q}$ with $\operatorname{Gal}(K/\...
- 91
9
votes
1
answer
519
views
Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$
I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows:
As a set $D=E\oplus uE \oplus u^2 E$ where $u$...
- 265
9
votes
2
answers
1k
views
Is it known if the absolute Galois group is "divisible"?
The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...
- 1,049
8
votes
1
answer
1k
views
Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \...
- 1,269
8
votes
3
answers
932
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$ ...
- 5,929
8
votes
0
answers
240
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 $Q(x_{1},x_{2},x_{3},...
- 3,645
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
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
1
answer
839
views
Incomplete Failures of the Inverse Galois Problem
I thought of this question the other day and have not been able to get any traction on references or results along its lines, so I finally caved and decided to ask it here. I am no expert on Galois ...
- 5,191
7
votes
2
answers
430
views
Do surface groups embed into PSL_2 over a real quadratic integer ring?
$\DeclareMathOperator\PSL{PSL}$ Let $ \mathbb{Z} $ be the ring of integers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma_g $ be a surface of genus $ g \geq 2 $. Let $ \pi_1(\Sigma_g) $ be ...
- 4,081
7
votes
1
answer
382
views
Inverse Galois problem for non-Galois extensions
The inverse Galois problem asks whether every finite group appears as the Galois group of a Galois extension of the rational numbers.
Is anything known about the anologous problem, where the ...
- 1,329
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}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
- 12.6k
7
votes
1
answer
603
views
Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
- 4,547
6
votes
5
answers
3k
views
Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real?
Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link ...
- 71
6
votes
1
answer
842
views
Is there a precise notion of "almost all" such that almost all finite groups are Galois groups of extensions of the rationals?
I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...
- 6,974
6
votes
2
answers
622
views
Inverse Galois problem for simple Lie type groups
Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in ...
- 17.6k
6
votes
1
answer
378
views
Enumeration of field generators of a finite field over $\mathbb{F}_{q}$, which are $m^{th}$ powers in the same field
Consider the finite field extension $\mathbb{F}_{{q}^{d}}$ over $\mathbb{F}_{q}$, where $q=p^a$ for some prime $p$. We assume $d\geq 2$. Let,
$$ S=\{ \alpha \in \mathbb{F}_{q^d}\hspace{0.1 cm} | \...
- 428
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
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
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
5
votes
3
answers
428
views
Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13?
I. Level 7
In Klein's "On the Order-Seven Transformations of Elliptic Functions", he gave two elegant resolvents of degrees 8 and 7 in pages 306 and 313. Translated to more understandable ...
- 12.6k
5
votes
2
answers
1k
views
non-continuous inverse Galois problem
Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$.
Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that
$G/K$ is free of ...
- 7,609
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
5
votes
1
answer
314
views
abelian and nonabelian parts of Aut($\widehat{F_2}$)
Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...
- 10.7k
4
votes
2
answers
1k
views
Unsolvability of a Quintic and its link with "Simplicity" of $A_{5}$
This is a re-post from MSE (because I did not get the kind of answer I wanted even after offering a bounty).
At the outset I must mention that I don't have a fairly working knowledge of Galois Theory ...
- 2,249
4
votes
1
answer
735
views
Shafarevich's theorem about solvable groups as Galois groups
I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.
- 101
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
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
4
votes
0
answers
245
views
Dessins d'enfants and the absolute Galois group
If I am not mistaken, Alexander Grothendieck introduced dessins d'enfants as they are known today, in order to better understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}...
- 4,547
4
votes
0
answers
124
views
Linear relation between polynomial roots
Consider an irreducible polynomial $P\in\mathbb{Q}[x]$ of degree $n$ whose second leading coefficient is $0$ and $\alpha_1,\dots,\alpha_n$ its $n$ distincts roots. I am interested on the problem of ...
- 231
3
votes
4
answers
2k
views
Explicit element in free group which is killed by every solvable quotient
The free group on two generators $F_2=\langle x,y|\rangle$ is the fundamental group of $\mathbb P^1(\mathbb C)\setminus\{0,1,\infty\}$. Now, there are plenty of galois covers of this space whose ...
- 18.7k
3
votes
1
answer
296
views
Distinct characters with the same character values, outer automorphisms and Galois conjugation
Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:
multiplying by a degree 1 character
applying an ...
- 4,081
3
votes
1
answer
379
views
How often does algebraic-conjugacy imply conjugacy?
Recall that two elements $h_1,h_2$ of a finite group $G$ are called conjugate when $h_2 = gh_1 g^{-1}$ for some $g \in G$, and algebraic-conjugate when $h_2 = gh_1^a g^{-1}$ for some $a \in (\mathbb{Z}...
- 54.6k