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
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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 $...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
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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 ...
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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.
...
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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 ...
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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, ...
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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 $n \geq 5$....
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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 ...
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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 ...
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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,...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 $8$ ...
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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 ...
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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 $\...
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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 ...
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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
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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 $t\mapsto t_0\in \mathbb{Q}$. ...
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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 ...
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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
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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 ...
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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},...
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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
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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 ...
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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
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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 $\mathbb{Q}(\zeta_{p^2})$ ...
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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 ...
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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 ...
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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
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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
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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
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511
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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 ...
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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 ...
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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
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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}} =...
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