Questions tagged [galois-groups]

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6 votes
0 answers
191 views

Is the group $\mathrm{Gal}(\mathbb{C}/\overline{\mathbb{Q}})$ known?

The automorphism group of the complex numbers $\mathbb{C}$ and the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are amongst the most mysterious and worst understood objects in Galois ...
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3 votes
0 answers
63 views

The cyclic analogue of the gonality of the superelliptic curve $s^n = t^m + 1$

For naturals $n$, $m > 1$ consider the superelliptic curve $C\!: s^n = t^m + 1$, for simplicity, over an algebraically closed field of zero characteristic or large characteristic $p \nmid n$, $m$. ...
0 votes
0 answers
111 views

Fixed space of absolutely Frobenius

Let $A$ be an affine space over $\bar{\mathbb{F}}_q$, $F$ be the absolutely Frobenius. Let $B$ be an $F-$ invariant affine subspace contained in $A$, $B^F$ be the fixed points of $F$ in $B$. My ...
2 votes
0 answers
97 views

Embeddings and images of number fields in $\mathbb{C}$ [closed]

Let $\mathbb{Q}(\alpha)$ be a number field, and suppose that $[\mathbb{Q}(\alpha) : \mathbb{Q} ] = m$. Then there are precisely $m$ different embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$, ...
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4 votes
1 answer
352 views

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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3 votes
1 answer
280 views

A criterion for the equation $ax^n+bx+c=0$ not solvable by radicals via $a,b,c$ and $n$

Galois revealed that an algebraic equation $f(x)=0$ with coefficients in a field $K$ of zero characteristic is solvable by radicals if and only if the Galois group of $f(x)$ over $K$ is solvable. ...
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4 votes
1 answer
264 views

Common Galois extension over $\mathbb Q $

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q} $ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...
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4 votes
0 answers
81 views

Index of norm $ 1 $ subgroup in a cyclic extension

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$...
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6 votes
0 answers
145 views

Galois group of zeta function of hyperelliptic curve

Let $f \in \mathbb F_q[T]$ be monic, squarefree. Can we say anything on the Galois group of $Z_f$, the zeta function of the hyperelliptic curve $y^2=f$, directly in terms of $f$ (coefficients or ...
0 votes
1 answer
144 views

Moving general fibers of a fibration

Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth. My ...
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5 votes
0 answers
163 views

Copies of the reals in $\mathbb{C}$ without the Axiom of Choice

Suppose we work in a model in which the Axiom of Choice does not hold, and in which $\mathbb{C}$ only has one nontrivial automorphism (such models exist). Question: "how many" subfields of $\...
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2 votes
0 answers
99 views

Permutation group with a nice lattice of block systems

Let $X$ be a finite set and $G$ be a transitive subgroup of the symmetric group on $X.$ Recall that a (complete) block system for this action is a partition of $X = B_1 \cup \cdots \cup B_k$ into ...
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6 votes
3 answers
735 views

Galois group of a polynomial modulo $p$

It is well known that if $f(x)$ is a polynomial over $\mathbb Z$ then for every prime $p$ (not dividing the discriminant of $f$ (thanks to KConrad)) the Galois group of that polynomial mod $p$ over $\...
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2 votes
0 answers
277 views

On the polynomials $x^n-x-1$ with $n>1$

In 1956, E. S. Selmer proved in a paper [Math. Scand. 4 (1956), 287-302] that for any integer $n>1$ the polynomial $x^n-x-1$ is irreducible over the field $\mathbb Q$ of rational numbers. Question. ...
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1 vote
1 answer
129 views

Galois extensions of ring spectra and subextensions

In Galois extensions of structured ring spectra, Rognes introduces the notion of a faithful $G$-Galois extension of ring spectra. Let me recall what this means: We have a commutative ring spectrum $R$ ...
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6 votes
1 answer
375 views

Perfect numbers, Galois groups and a polynomial

Let $f(n,t) = \sum_{k=0}^{r-1} d_k t^k$ where $D_n = \{d_0=1,d_1,\cdots,d_{r-1}\}$ are all divisors of $n$. For instance $$f(28,t) = 28 t^{5} + 14 t^{4} + 7 t^{3} + 4 t^{2} + 2 t + 1$$ For even ...
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3 votes
0 answers
163 views

Atlas of polynomial Galois groups

A polynomial $p \in \mathbb{Z}[x]$ of degree $n$ can be encoded as a finite sequence $(a_0,a_1, \dots, a_n)$, i.e. $p(x)= \sum_{i=0}^n a_i x^i$. Let $G(a_0,a_1, \dots, a_n)$ be the Galois group of the ...
3 votes
0 answers
127 views

Explicit construction of abelian wild inertial extensions of maximal tamely ramified extension of $\mathbb{Q}_p$?

In Iwasawa's paper On Galois groups of local fields, he proves that if $V$ is the maximal tamely ramified extension of $\mathbb{Q}_p$, with Galois group $\Gamma$ over the base, then its abelianized ...
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0 votes
2 answers
444 views

Splitting field of an intermediate field

Consider the following 'wrong' question. Let $f(x) \in F[x]$ be an irreducible polynomial in a polynomial ring of a field $F$. Let $L$ be the splitting field of $f(x)$ over $F$. Assume that $L$ is a ...
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9 votes
0 answers
369 views

Does local Langlands say anything about the isomorphism class of the absolute Galois group?

I have heard some people claim that the Local Langlands Correspondence over $\mathbb{Q}_p$ (when it is known) is a deep theorem about representations of the absolute Galois group of $\mathbb{Q}_p$. My ...
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6 votes
1 answer
531 views

Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
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2 votes
0 answers
180 views

Closed points and the absolute Galois group

Suppose $\chi$ is a scheme of finite type over the field $k$; define $\overline{\chi} := \chi \otimes_{\mathrm{Spec}(k)} \mathrm{Spec}(\overline{k})$, with $\overline{k}$ an algebraic closure of $k$. ...
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6 votes
0 answers
176 views

Variation of Galois groups of a cover in families

Once again sorry for the formatting, I'm using a phone. Fix an étale cover of $Y\times S$, where $S$ is connected. We pull-back along inclusions of points into $S$ to get a family of étale covers of $...
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10 votes
0 answers
283 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 $...
1 vote
1 answer
265 views

What does the group of automorphisms corresponding to $\mathfrak{g}$

I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base ...
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10 votes
1 answer
432 views

The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$

I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$: Is the double cover of $Sp_6(...
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9 votes
1 answer
405 views

Using Jordan's theorem to find Galois group for a polynomial

I'm trying to apply the result of Jordan's theorem (cited below) to find the Galois group for a given polynomial. My goal is to provide an example where Jordan's theorem is useful, so the polynomial I'...
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5 votes
0 answers
96 views

Statistics for Galois groups of polynomials of bounded height

Let $f(x)=\sum_{i=0}^ma_ix^i$ be a monic polynomial with integer coefficients, and let $H(f)=\max_i\{|a_i|\}$. Moreover, let $\text{Gal}(f)$ denote the Galois group of $f$. Now, for every positive ...
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1 vote
1 answer
114 views

Bounding the number of polynomials whose Galois group is a subgroup of the alternating group

In the article of DAVID ZYWINA https://pdfs.semanticscholar.org/cd50/c2d3fb0ce0c6a66ee629419b69165b30d5bc.pdf. It says that using $n$- dimensional large sieve, we can get the bound |$\{$$ \ \ f(x)=x^...
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10 votes
1 answer
852 views

Atiyah on the "Galois group of the octonions" and Physics

Apparently Atiyah was talking about the "Galois group of the octonions" and the unification of the forces of physics at the Heidelberg Forum. Unfortunately not on the stage -- it didn't make its way ...
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10 votes
1 answer
1k views

Which of these 4 definitions of Galois coverings of integral schemes are equivalent?

Here are four possible definitions for an etale, finite, surjective map $X\rightarrow Y$ between integral schemes to be considered Galois: There exists a finite group $G$, and an action $\varphi: G\...
2 votes
0 answers
328 views

Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers. Question 1: Is it true that $\mathbf{C}$ is ...
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8 votes
1 answer
677 views

Question on Inverse Galois Theory

Let $G$ be a finite group, $n=|G|$. Let $\rho:G\rightarrow GL(n,\mathbb{C})$ be the regular representation. Let $G \le H \le S_n$ be another group. Then we have $\mathbb{Q}[x_1,\cdots,x_n]^H \le \...
user avatar
1 vote
2 answers
265 views

Generic polynomial for alternating group ${A}_{4}$ is not correct

I was validating the percentage of cases where the generic two parameter polynomial for Galois group ${A}_{4}$ is valid. We have \begin{equation*} {f}^{{A}_{4}} \left({x, \alpha, \beta}\right) = {x}...
5 votes
0 answers
113 views

Explicit extensions for Heisenberg groups

Let $G$ be the $p$-adic Heisenberg group $\begin{pmatrix} 1&\mathbb Z_p&\mathbb Z_p\\&1&\mathbb Z_p\\&&1\end{pmatrix}$. Is it possible to write an explicit extension $K/k$, ...
2 votes
0 answers
285 views

Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...
3 votes
0 answers
86 views

Finding a divisor on a curve in a given linear equivalence class made with points over a "small" field

Suppose I have a curve $C$ defined over $\mathbb Q$ and a line bundle $\mathcal L$ on it, also defined over $\mathbb Q$. I am interested in trying to find a (non-effective) divisor $D=\sum_i a_ip_i$ ...
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5 votes
1 answer
325 views

Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$. Is it true that ...
9 votes
2 answers
1k views

Hodge structure versus Weight structure

This is a naive question. One is told that, somehow, Hodge theory for varieties over complex numbers, is an analog of weight theory for varities over finite fields. In weight theory, one considers ...
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4 votes
1 answer
438 views

Does there exist an order in a number field of deg>1 with a map to F_p for all p?

This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that ...
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0 votes
1 answer
601 views

Does this isomorphism between Galois groups hold for transcendental extensions?

Suppose $L/K$ and $M/K$ are algebraic extensions of a field $K$, such that $L\cap M=K$, and $L/K$ is a normal extension. It is well-known that, with these conditions, we have: $$\text{Gal}(L/K)\cong ...
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9 votes
1 answer
3k views

Maximal tamely ramified extension of $\mathbf Q_p$

Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?
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6 votes
3 answers
1k views

Solving $z^n=a+bi$ using only radicals of positive real numbers

Let $a+bi\in\mathbf{C}$ be a complex number with $a,b\in\mathbf{R}$. Then it is easy to find exact solutions of $z^2=a+ib$. For example let $z=u+iv$. Then $$ u^2+v^2=z\overline{z}=\sqrt{a^2+b^2} $$ ...
3 votes
2 answers
1k views

Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?

If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
1 vote
0 answers
250 views

Automorphism groups of fields

Hi there, is there a classification/characterization of fields K for which the automorphism group Aut(K) has the property that |Aut(K)| < |K| (e.g. finite fields, the rationals and reals) ? What ...
  • 3,589
3 votes
2 answers
386 views

Subject to some conditions, is it possible to conclude a subfield of an abelian extension generated by a unit is a cyclic extension

My research is mostly in the area of modular categories. In the course of my research I came across a constraining set of number theoretic conditions that I'd like to exploit. It has been pointed out ...
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1 vote
1 answer
1k views

What does Gal(Q_p/Q) mean? [closed]

What does $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.) If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$, then does it have any property ...
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11 votes
1 answer
1k views

Extensions obtained adding torsion points of an elliptic curve

When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, \mathbb{...
8 votes
2 answers
762 views

Polynomial with Galois Group $D_{2n}$

How does one construct a polynomial with Galois Group $D_{2n}$? A general method would be preferable or if that's impractical then an example of it being done for any n would be appreciated. Thanks!
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26 votes
3 answers
2k views

Galois Groups of a family of polynomials

I've stumbled across the family of polynomials $ f_p(x) = x^{p-1} + 2 x^{p-2} + \cdots + (p-1) x + p $, where $p$ is an odd prime. It's not too hard to show that $f_p(x)$ is irreducible over $\mathbb{...