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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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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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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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 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,
$$j=\...
- 2,959
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Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
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Galois action on blow-ups related to field extensions of infinite degree
Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...
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When does the set of Frobenius conjugacy classes happen to be the whole infinite Galois group?
Let $ K$ be a number field and let $S$ be a finite set of places that contains the archimedean places. Let $G_{K,S}=\operatorname{Gal}(K_{S}/K)$ be the Galois group for a maximal extension $K_S/K$ ...
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On reducing degree-$12$ equations with Mathieu group $M_{11}$ to its degree-$11$ resolvent?
The general $4$-deg and some $8$-deg (such as the Schein octic) when a linear transformation is done so their $x^{n-1}$ term vanishes can have a neat solution as,
$$x = \sqrt{z_1}+\sqrt{z_2}+\sqrt{z_3}...
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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,
$$x^7-7x^4-14x^3-7=0$$
$$x^8+x^7+29x^2+29=0$$
$$x^9-27x^4-9x^3-9^2=0$$
$$x^{12}-36x^5-12x^3-...
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Learning Inverse Galois Theory
Can someone give me a roadmap for learning Inverse Galois theory?
I am a PhD student in the representation theory of finite groups. I studied Galois theory when I was an undergraduate student. The ...
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Decomposition groups for the Galois module $\mu_8$
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\...
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Is there an irreducible but solvable septic trinomial $x^7+ax^n+b = 0$?
The following irreducible trinomials are solvable:
$$x^5-5x^2-3 = 0$$
$$x^6+3x+3 = 0$$
$$x^8-5x-5=0$$
Their Galois groups are isomorphic to ${\rm D}_5$, ${\rm S}_3 \wr {\rm C}_2$ and
$({\rm S}_4 \...
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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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“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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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 ...
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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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159
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A dimension problem related to an abelian simple extension of a field
$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
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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....
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Why do we make such big deal about the 'unsolvability' of the quintic?
The unsolvability of a general quintic equation in terms of the basic arithmetic operations and $n$th roots (i.e. the Abel–Ruffini theorem) is considered a major result in the mathematical canon. I ...
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Maximal abelian subgroups of absolute Galois group
For both local and global fields, we have a good handle on the abelianization of the absolute Galois group of $K$. Essentially this allows us to "understand" all maps from $G_K$ to abelian ...
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Infinite tamely ramified $p$-extensions of $\mathbb{Q}$ contain infinite unramified subextensions?
Let $p$ be a prime. By a $ p $-extension we mean a Galois extension whose Galois group is a $ p $-group. Let $L$ be an infinite tamely ramified $p$-extension of $\mathbb{Q}$, i.e. all primes ramified ...
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Using the Lehmer quintic to solve $11$-degree equations and higher?
(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
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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 $L$ (...
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A similar relationship between the generic cubic and the Lehmer quintic?
I. Comparison
It doesn't seem to be well-known that the generic cubic (prominent in this MO post) for $C_3 = A_3$,
$$x^3-nx^2+(n-3)x+1 = 0$$
has the nice property that its roots $a,b,c$, if in correct ...
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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 $...
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Solving solvable septics using only cubics?
After the satisfying resolution of my question on the Kondo-Brumer quintic, I decided to revisit my old post on septic equations.
I. Solution by eta quotients
The septic mentioned in that post may not ...
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On the refined minimal ramification problem for $p$-groups
Let $p$ be a prime. The minimal ramification problem is to ask whether or not every finite $p$-group $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly ...
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On V. Arnold's trinities regarding PSL(2,5), PSL(2,7), and PSL(2,11)?
Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}.$ Then the following functions for levels $...
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Relation between $G_{\mathbb{Q}_p}$ for different primes
Let $G_{\mathbb{Q}_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}_{p}$. Also, their structure as abstract groups is completely known.
It is well known that this group embeds ...
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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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Finite normal extensions
Suppose that $K$ is a finite field extension of $F$. Is the following equivalent to the extension being normal?
If $L$ is an extension of $K$ and $\sigma:K\to L$ fixes $F$, then $\sigma(K) = K$.
I ...
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Construction of a symmetric polynomial in the roots that acts like the discriminant
The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...
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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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Ramifications in Galois closures of number fields
Let $K/\mathbb{Q}$ be a finite Galois extension, and $L/K$ an infinite Galois extension such that the Galois group $ \operatorname{Gal}(L/K) $ is isomorphic to a closed subgroup of $ \operatorname{GL}...
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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 $\...
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Quadratic extension of local field
Let $F$ be a nonarchimedean local field of characteristic zero, and $E$ an extension of $F$ with $[E:F]=2^n$ for some $n$. Is it always possible to find a quadratic extension $M$ of $F$ such that $F\...
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On the solvable octic $x^8-x^7+29x^2+29=0$
The irreducible but solvable octic,
$$x^8-x^7+29x^2+29=0\tag{1}\label{1}$$
was first mentioned by Igor Schein in this 1999 sci.math post (Wayback Machine). This does not factor over a quadratic or ...
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0
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Projection map $\pi:\left(\mathcal{O}/n\mathcal{O}\right)^\times \to\left(\mathcal{O}/\gcd(n,m)\mathcal{O}\right)^{\times}$ of a CM elliptic curve
In this paper the author has mentioned in page $693$ under section $2.2$ that for an Elliptic curve $E/\mathbb{Q}$ with CM by an order $\mathcal{O}$ of an imaginary quadratic field $K$ there is a ...
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1
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Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups
Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
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1
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A simple proof of the fundamental theorem of Galois theory
Update. It's now on the arXiv.
Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite ...
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1
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Etale cohomology and Kummer theory
If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
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1
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A question about unramified quadratic extension of number field
Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?
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Can we say anything about the zeros and Galois group of the polynomial $(x^p-a)^{p^2}-p^{p^2+1}x+p^{p^2} a=0$?
Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$.
Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...
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Is there a theory of "elementary closed form solution" at the operator level for differential equations?
We begin by considering the usual general first order linear equation of the form
$$ a_0 y' + a_1 y + a_2 = 0 $$
Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
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When is $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ a Galois extension of $\mathbb{Q}$?
When is $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ a Galois extension of $\mathbb{Q}$?
I was motivated by the question that $\mathbb{Q}(\sqrt{5+\sqrt{5}})$ is a Galois extension of $\mathbb{Q}$. Here is a rough ...
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1
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What are the jumps in the ramification filtration of the absolute Galois group of a local field?
Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...
- 148k
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Galois cohomology with coefficients in the integers of the Lubin-Tate extension
Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
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