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.
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The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?
I asked this question over a year ago on Math.StackExchange but I didn't get an answer.
In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
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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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Galois groups associated to matrices
When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even.
Question: For every $p$, does ...
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Characteristic polynomials of Cartan matrices of Lie algebras
Let $L$ be a simple Lie algebra over $\mathbb{C}$ with Cartan matrix $C_L$ (as in https://de.wikipedia.org/wiki/Cartan-Matrix )
Question 1: Is is true that the characteristic polynomial $f$ of $C_L$ ...
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$SL_2(\mathbb{Z}_p)$ extension of a local field
Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...
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How to (easily) obtain the splitting field for dihedral extensions
Let $f(x) \in {\mathbb Q}[x]$ be a polynomial that is irreducible over ${\mathbb Q}$ with $D_{n}$ (the dihedral group of order $2n$) as its Galois group. Let $\alpha$ be a root of $f(x)$ and put $K={\...
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Galois action on Grothendieck ring of varieties
Let $k$ be a field and let $\overline{k}$ be its algebraic closure. Let $K_0(Var/\overline{k})$ be the Grothendieck ring of algebraic varieties over $\overline{k}$.
Is it true that the natural ...
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Criteria for irreducibility using the location of complex roots
I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
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Correspondences of schemes induced by a finite Galois extension
I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes.
Recall that if $...
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Absolute Galois group, number theory and the Axiom of Choice
Richard Taylor once explained his research (in number theory) in a very simple way: understanding the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
It is known that in ...
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Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?
I asked this question at MSE but I did not receive an answer. So I ask it at MO:
We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by $Gal(f)$...
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Disjoint images of polynomials
Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
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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) \simeq\Gal(E/\...
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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_{...
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A special integral polynomial
Given $n \in \mathbf{N}$,is always possible to construct a monic polynomial in $\mathbf{Z}[x]$ of degree $2n$, whose roots are in $\mathbf{C} \setminus \mathbf{R}$ and whose Galois group over $\mathbf{...
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Octic family with Galois group of order 1344?
Does the octic,
$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$
for any constant n have Galois group of order 1344? Its discriminant D is a perfect square,
$D = (1728n^4-341901n^3-...
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Rationality of field embeddings
After my earlier question question turned out to have a negative answer (Thank you to all respondents!), here is a more modest one. Both a positive answer and a counterexample would help my work. If ...
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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 ...
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Cycle type in Galois group from ramified primes
Let $P \in \mathbb Z[X]$ be monic, separable, of degree $d$, $K$ its splitting field over $\mathbb Q$ and $G$ the Galois group of $K$ over $\mathbb Q$.
Now, let $p$ be a prime number unramified in $K$....
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Embeddings of $\overline{\mathbf{Q}}$ into $\mathbf{C}$
Keenan Kidwell's answer to Place stabilizers for the absolute Galois Group mentions that "choosing a complex conjugation" in $G_{\mathbf{Q}}$ means choosing an embedding $\overline{\mathbf{Q}}\...
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Constructive proof of algebraic elements forming a subfield
Let $F \leqslant E$ be a field extension.
If $a, b \in E$ are algebraic over $F$ then $a+b$ and $ab$ are algebraic as well. There is an short proof of this using the extension $E(a,b)$:
$[E(a,b):E]$ ...
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Effective Chebotarev density results for arbitrary number fields
So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define
the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of
the ...
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Galois action on algebraic K-theory of finite fields
This might be well-known to experts. I was just teaching a course where we went through some parts of Quillen's theorem computing the higher algebraic K-theory of finite fields. Denote by $\mathbb F_q$...
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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 ...
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On the solvability of the equation $ax^p+bx^{p-1}+cx+d=0$ by radicals
Joint with Qing-Hu Hou at Tianjin Univ., we seek for explicit criteria via coefficients for the solvability of an algebraic equation by radicals. In this direction, we formulate the following ...
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Is there a measurable isomorphism ${\mathbb C}\to {\mathbb C}_p$?
Let $p$ be a prime and ${\mathbb C}_p$ be the completion of the algebraic closure $\overline{{\mathbb Q}_p}$. This field is isomorphic to $\mathbb C$. Both fields come with natural absolute values but ...
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Is it always possible to "separate" the eigenvalues of an integer matrix?
Call a square matrix Galois-irreducible if all its eigenvalues are Galois conjugates of each other.
Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always possible ...
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What is the relation of the absolute Galois group and classical profinite groups?
Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G_p = \mathrm{Gal}(\overline{\mathbb{Q}_p}: \mathbb{Q}_p)$.
Abelian class field theory gives us for the ...
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Splitting of a division algebra with an involution of second kind
Let $k$ be a field, $K/k$ a separable quadratic extension,
and $D/K$ a central division algebra of dimension $r^2$ over $K$
with an involution $\sigma$ of second kind
(i.e. $\sigma$ acts non-trivially ...
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Restricting maps between strict henselisations
$\require{AMScd}$I am currently thinking about (strict) henselisations but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict ...
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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 ...
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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 ...
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Is it possible to solve sextic equations using the Fox H function?
Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago.
In contrast, we know more about the Fox H function, and we ...
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Dihedral extension unramified at primes dividing order of group?
Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $...
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Criterion for generic polynomials
Generic polynomials, which are recalled below, play an important role in the constructive aspects of the inverse Galois problem.
Definition. Let $P(\mathbf{t},X)$ be a monic polynomial in $\mathbb{Q}...
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Which quartic fields contain the 4th roots of unity in their Galois closure?
Let $K/\mathbb{Q}$ be a degree $4$ number field. Is it known how to determine whether the Galois closure of $K$, say $K'$, contains the 4-th roots of unity?
In the cubic case, there is a succinct ...
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Exceptional specializations of Galois groups in the Hilbert Irreducibility Theorem
Suppose $f(x,t)\in\mathbb{Q}(t)[x]$ is an irreducible polynomial with Galois group G. For any rational number $a$ we may consider the polynomial $f(x,a)\in\mathbb Q[x]$ and its corresponding Galois ...
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Infinite Galois equivalence
I am having some trouble trying to understand the proof of Theorem 7.2.5 in Bhatt and Scholze's paper The pro-étale topology for schemes. Specifically, I don't quite understand why it was necessary to ...
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Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?
I. Kondo-Brumer quintic
The deceptively simple solvable quintic,
$$x^5 + (a - 3)x^4 + (-a + b + 3)x^3 + (a^2 - a - 1 - 2b)x^2 + b x + a=0$$
is quite important for imaginary quadratic fields. For ...
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Frobenius eigenvalues algebraic numbers
Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...
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Correspondence between coverings and field extensions
I am self reading from Groups as Galois Group by Helmut Volklein
There is a result on page 94(section 5.4)
Let $G$ be a finite group. Let $P\subset P^{1}$ finite and $q\in P^{1}\P$. There is a ...
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Can completely multiplicative functions be extended to $\overline{\mathbb{Q}}$ or further?
I'm looking for a subject of study that handles the following question. I'm not the most familiar with algebra; I have a strong working knowledge and that's about it, but I've been considering ...
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Are all rational exactly solvable differential equations known?
Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...
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Properties of Mod $\ell^m$ Galois representation associated to modular form
(Sorry for my poor english..)
Let $F(z)\in S_{2k}(SL_2(\mathbb{Z})$) be a newform and $\ell$ be a prime larger than $3$. Let $K$ be a some number field and $v$ be a prime of $K$ over $\ell$. Let $K_v$...
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Elliptic curves and $GL(2)$ Iwasawa theory
Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...
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Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?
If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute ...
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Is there $t\in\operatorname{Gal}(\overline{K}/K)$ s.t. $\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$?
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$
and $G:=\operatorname{Gal}(\overline{...
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Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
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Field of Definition of a Meromorphic Function
Question
Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field,...
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“Sheaf cohomology” of Galois groups
Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see ...