Questions tagged [galois-theory]
Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.
263 questions with no upvoted or accepted answers
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
30
votes
0
answers
2k
views
What do dessins tell us about the absolute Galois group?
I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single ...
- 40.2k
28
votes
0
answers
907
views
On certain representations of algebraic numbers in terms of trigonometric functions
Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...
- 6,819
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
17
votes
0
answers
750
views
Elements of finite fields with many powers of trace zero
Let $p$ be an odd prime number, $n>1$ be an integer, and $\mathrm{tr}$ be the trace map of the field extension $\mathrm{GF}(p^{2n})/\mathrm{GF}(p)$. For which pair $(p,n)$ does there exists $x\in\...
- 767
16
votes
0
answers
531
views
Aligned roots of irreducible polynomials
It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
- 13.4k
16
votes
0
answers
1k
views
Galois theory timeline (II)
This question is a sequel. I structured the previous one around Emil Artin's classic treatment of Galois theory from the 1940s, though making clear some reservations of my own about whether Artin ...
- 12.6k
15
votes
0
answers
458
views
Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
- 11.3k
15
votes
0
answers
866
views
On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$
(Major revision to incorporate new results in this MO cubic version.)
Note: All coefficients are in the rationals.
I. Cubic
In the linked post, it was shown that given a general cubic (via its ...
- 12.6k
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
0
answers
376
views
What was the "stormy discussion" about differential Galois theory at IHES?
In Kazuo Okamoto and Yousuke Ohyama's paper "Mathematical works of Hiroshi Umemura", Annales de la faculté des sciences de Toulouse Mathématiques, XXIX, no. 5 (2020) pp. 1053-1062, there is ...
- 1,826
13
votes
0
answers
624
views
On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?
(Updated with new information.)
I. Five eta quotients and the Monster?
Given Dedekind eta function $\eta(\tau)$, define the five eta quotients which in fact are the McKay-Thompson series 1A, 2A, 3A, ...
- 12.6k
13
votes
0
answers
247
views
Galois groups of special polynomials
This question is motivated by long experiments with GAP.
Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
- 26.5k
13
votes
0
answers
243
views
Galois group of polynomials related to Fibonacci and Catalan numbers
Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers.
Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$.
For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$.
And another ...
- 26.5k
13
votes
0
answers
287
views
Actions of $\mathbb Z/2\mathbb Z$ on algebraically closed fields and even-dimensional spheres and parallel between Galois theory and covering theory
It is well known that there is a parallel between Galois theory and covering theory. So I wonder whether there is a deep similarity between the following two facts:
Artin-Schreier theorem. The only ...
- 1,980
12
votes
0
answers
325
views
Is there an algorithm to compute a Belyi map for the Riemann surface?
Let $y^2=x^5-x-1$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $\{0,...
- 269
12
votes
0
answers
265
views
Galois groups of classical differential equations
I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following:
Is there a book or article devoted (either partially or completely) to ...
- 4,746
12
votes
0
answers
3k
views
Algebraic Closure of the field of rational functions
Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is $\...
- 1,891
11
votes
0
answers
676
views
Evaluating products of cyclotomic polynomials at roots of unity
Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...
- 19.7k
11
votes
0
answers
1k
views
Do we know that 'most' finite groups are Galois groups of number fields?
The inverse Galois problem is a classical problem in mathematics and asks whether every finite group can be realized as the Galois group of a finite field extension of the rational numbers. The ...
- 26.9k
10
votes
0
answers
234
views
If $H$ is a quotient of $G$, does there exist an $H$-extension of $\mathbb{Q}$ not contained in a $G$-extension?
Let $\phi\colon G\rightarrow H$ be a surjective homomorphism between finite groups. Assume that $\phi$ is not split, in other words there exists no homomorphism $\sigma\colon H\rightarrow G$ such that ...
- 984
10
votes
0
answers
452
views
What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
- 32.5k
10
votes
0
answers
379
views
Recognizing the Galois group from the field discriminant
Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...
- 2,959
10
votes
0
answers
131
views
Inverse Galois problem for $2$-groups with an involution as complex conjugation
It is known that the inverse Galois problem for solvable groups was solved by Shafarevich. My question is the following: given $G$ a finite $2$-group and $s$ an element of order $2$ in $G$. Can we ...
- 715
10
votes
0
answers
308
views
Are polynomials with non-($S_n$ or $A_n$) Galois groups discrete?
There are various results, beginning with van der Waerden, on the asymptotic number of integral polynomials with bounded coefficients, whose Galois group is $S_n$. Some of these results also include $...
- 201
10
votes
0
answers
287
views
Galois action on units of totally real cyclic numberfields
Given a cyclic totally real number field $K/\mathbb{Q}$ of degree n with unit group isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}^{n-1}$, how much is known about the action of Gal$(K/\mathbb{Q})$ on ...
10
votes
0
answers
248
views
Expressing every algebraic number using roots of trinomials?
This question is a continuation of Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...
- 11.3k
10
votes
0
answers
713
views
The construction of the 257gon
If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...
- 47.3k
10
votes
0
answers
1k
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 ...
- 1,288
9
votes
0
answers
261
views
Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
- 41.9k
9
votes
0
answers
256
views
Compass and straightedge construction of Poncelet polygons
Gauss–Wantzel theorem states that
A regular n-gon is constructible with straightedge and compass if and only if $n = 2^kp_1p_2...p_t$, where $p_i$'s are distinct Fermat primes (A Fermat prime is a ...
- 691
9
votes
0
answers
699
views
Motivic Galois theory and Betti realizations?
Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
- 99
9
votes
0
answers
203
views
Characterizing regular Galois extensions by the set of their specializations
Let $E$ and $F$ be two regular Galois extensions of $\mathbb{Q}(t)$ with group $G$, and let $E_{t_0}$ resp. $F_{t_0}$ be the residue fields corresponding to specializing $t\mapsto t_0\in \mathbb{Q}$. ...
- 91
8
votes
0
answers
294
views
Relationships among constructions of fundamental group for schemes
There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
- 447
8
votes
0
answers
314
views
Finding a cyclic cubic extension of a field
Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...
- 1,017
8
votes
0
answers
376
views
Cutting and pasting in Galois theory
I want to ask who was the first to use cut-paste construction in Galois theory.
This question is motivated from the trend in contemporary Galois theory to use patching methods to construct Galois ...
- 3,362
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
0
answers
157
views
Non-abelian ray class fields for local fields
Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
- 1,116
7
votes
0
answers
149
views
Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
- 299
7
votes
0
answers
205
views
Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$
$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
7
votes
1
answer
707
views
How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
- 707
7
votes
0
answers
391
views
Do algebraic completion/topological completion of fields always terminate? If so, are they unique?
Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$.
On the other hand, the ...
- 171
7
votes
0
answers
252
views
Possible Galois groups of residue fields of ramification points of a Galois branched cover of curves
Here's the version of the question I'm particularly interested in (though I'm also interested in other variants described near the end).
Let $K$ be the function field of a smooth projective variety ...
- 10.7k
7
votes
0
answers
330
views
Non-linear Galois descent
This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...
- 1,153
7
votes
0
answers
253
views
Is this Related to Tannakian Formalism?
I am wondering how I might be able to express the following phenomenon, which is essentially equivalent to Artin's linear independence of characters, in Tannakian formalism. Any help would be much ...
user30211
6
votes
0
answers
169
views
Classification of Étale algebras without Galois theory and then deducing Galois theory
In Milne's Galois theory notes — chapter 8, quoted below, he remarks that it is possible to classify étale algebras without using Galois theory then deduce Galois theory and he will explain this ...
- 253
6
votes
0
answers
511
views
Extensions of p-adic number fields
Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...
- 71
6
votes
0
answers
271
views
What, if anything, do we hope and expect to understand about (classical) Galois groups?
I was reading Franz Lemmermeyer's introduction to Fermat's Last and Wiles' Theorem, where he states
Galois representations $\rho_p : G_\mathbb Q\rightarrow GL_2(\mathbb Z_p)$ are used for studying ...
- 982
6
votes
0
answers
421
views
Can Langlands correpondence be restated using topos?
Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions.
Laurent Lafforgue applying Olivia Caramello thesis described in ...
- 77
6
votes
0
answers
147
views
Using the Rogers-Selberg identities to solve certain septics?
Given the Ramanujan theta function $f(a,b)$ and the Rogers-Selberg identities,
\begin{align}
U_1 &= \frac{f(-q,-q^6)}{f(-q^2)} = \sum_{n=0}^\infty \frac {q^{2n^2+2n}} {(q^2;q^2)_n\,(-q;q)_{2n+1}} =...
- 12.6k