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.
850 questions
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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) ...
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Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$
Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
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For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$
Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
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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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Absolute Galois cohomology of function fields (of high-dimensional) varieties
What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$.
I have seen ...
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Existence of a symmetric matrix satisfying certain irreducible conditions
Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
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Galois connection for homeomorphisms
let $M = \mathbb{R^2}$ and $X = \{0\}$ and $G = Aut_X(M)$ the group of homeorphisms fixing $X$ (pointwise). Then we have, in analogy to classical Galois theory for field extensions, a Galois ...
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Global class field theory and closure of unit groups
I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
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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
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Is there work on differential Galois theory and infinite operators?
I'm curious about differential Galois theory and I've noticed that everything I read covers only finite order operators (e.g. $L = Y^n + a_{n-1} Y^{n-1} + \dots + a_0 Y$). Has there been any work on ...
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A cyclic Galois extension over $ \mathbb{Q}(\omega)$
It is known that $\mathbb{Q}(\sqrt{-1})$ does not live in a cyclic Galois extension $L$ of $\mathbb{Q}$ of degree $4$. For example, the image of complex conjugation in $\mathrm{Gal}(L/\mathbb{Q}) = \...
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Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$
Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
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Chinese remainder theorem for composition
Let $f(x) \in F_p[x]$ and I know $f(x)$ modulo two polynomials $\phi_1(x)$ and $\phi_2(x)$. What sort of information about $f$ modulo the composition $\phi_1(\phi_2(x))$ can I recover?
I'm looking ...
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E=F(E^p) implies extension is algebraic
Let $F$ be a field of characteristic $p$ and $E/F$ be a finitely generated field extension such that $E=F(E^p)$. Then show that $E/F$ is algebraic.
I have proven it in case $E$ is singly generated ...
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If degree $N$ polynomials always have a root, does there still exist an irreducible polynomial of degree $N+1$?
Given that $F$ is a field, let $F_n$ be the completion of $F$ with respect to roots of degree $n$ polynomials. For example this would make $\mathbb{Q}_2$ the field of (ruler and compass) constructible ...
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Unramified extension over $ \mathbb{Q}_{p} $
Let $\mathbb{Q}_{p}$ be a p-adic field such that $ p \neq 2 $. We knew that for every $ n=2m $ there exists exactly one unramified extension $ K $ of $ \mathbb{Q}_{p} $ of degree $ n $, obtained by ...
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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 ...
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Good and bad reduction for twists of algebraic curves
Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$.
Suppose that $C$ has good reduction at a ...
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Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
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irreducibility of the polynomial $ x^4 +1 $
Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
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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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Fields in which $ -1 $ can't be written as sum of two square elements
We say a field $F$ has the property $*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $*$. On the other hand if $ F $ is ...
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Statistics of action of Galois group of number field on primes over unramified rational primes
Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}_K$ over $p$. Do we know any statistical information ...
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A variant of the inverse Galois problem
In Theorem I of Construction of maximal unramified p-extensions with prescribed Galois groups, it's proved that
for any prime $p$ and any given finite $p$-group $G$, there exists a number field $F$ ...
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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}...
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Algebraically closed fields with only finite orbits
The automorphism groups $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ of algebraic closures of finite fields $\mathbb{F}_q$ and the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\...
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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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Not containing $ 2^{i}$-th primitive roots of unity in cyclic galois extension of number field
Suppose $K$ is an algebraic number field. We want to determine whether, for all even $n$, there always exists a cyclic Galois extension $L$ of degree $n$ over $K$ such that the intermediate field $K_2$...
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Norm of $2^{i}$-th primitive root
Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ ...
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Splitting fields of degree 4 irreducible polynomials containing a fixed quadratic extension
Let $f_1(x)\in \mathbb{Z}[x]$ be a fixed irreducible degree 4 polynomial such that its splitting field $F_1$ is an $S_4$-Galois extension over $\mathbb{Q}$ and the discriminant of $F_1$ is of the form ...
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Is a generic genus $g \geq 7$ curve a solvable cover of $\mathbb{P}^1$?
Let $Y \to X$ be a finite branched cover of smooth projective curves over $\mathbb{C}$, so we get a finite extension $K(Y)/K(X)$ where $K(\ )$ is the field of meromorphic functions. Say that $Y \to X$ ...
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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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Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$
I. Degree 8
Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$
\begin{align}
{j_1}\; &=\frac{(x^2 + ...
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On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas
I. First Set
Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have,
$$j=\...
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Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13?
I. Level 7
In Klein's "On the Order-Seven Transformations of Elliptic Functions", he gave two elegant resolvents of degrees 8 and 7 in pages 306 and 313. Translated to more understandable ...
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Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
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Galois action on blow-ups related to field extensions of infinite degree
Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...
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When does the set of Frobenius conjugacy classes happen to be the whole infinite Galois group?
Let $ K$ be a number field and let $S$ be a finite set of places that contains the archimedean places. Let $G_{K,S}=\operatorname{Gal}(K_{S}/K)$ be the Galois group for a maximal extension $K_S/K$ ...
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On reducing degree-$12$ equations with Mathieu group $M_{11}$ to its degree-$11$ resolvent?
The general $4$-deg and some $8$-deg (such as the Schein octic) when a linear transformation is done so their $x^{n-1}$ term vanishes can have a neat solution as,
$$x = \sqrt{z_1}+\sqrt{z_2}+\sqrt{z_3}...
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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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Decomposition groups for the Galois module $\mu_8$
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\...
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1
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On the solvable septic quadrinomial $x^7-7x^4-14x^3-7=0$?
The concept of sparse polynomials has its place, and solvable but irreducible quadrinomial examples such as,
$$x^7-7x^4-14x^3-7=0$$
$$x^8+x^7+29x^2+29=0$$
$$x^9-27x^4-9x^3-9^2=0$$
$$x^{12}-36x^5-12x^3-...
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1
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What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
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6
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2k
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Can the positive root of this polynomial be expressed elementarily?
For each real $A>0$, let $x_A$ denote the positive root $x$ of the polynomial $x^5-3x-A$. Is the function $(0,\infty)\ni A\mapsto x_A$ elementary?
[I am using this definition of elementary ...
- 128k
5
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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 ...
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1
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423
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How do I extend the $2$-adic absolute value to prove Monsky's Theorem?
In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user500557
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$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$
Lets start with a definition.
Rules ...
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Algebraic numbers with a polynomial property
In my research I faced with an intricate construction of an algebraic number with some properties.
Problem. For which classes of polynomials $P(X,Y)\in \mathbb{Z}[X,Y]$, we have the following property....
- 515
3
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3
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Irreducibility of polynomials over some number fields
Recently, a problem in my research has appeared and now I need to construct some algebraic numbers with special properties (related to its degree and some other fields extensions).
Now, in order to ...
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