# Questions tagged [galois-groups]

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51
questions

6
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191
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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 ...

3
votes

0
answers

63
views

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

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
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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 $\mathbb{C}$, ...

4
votes

1
answer

352
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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 ...

3
votes

1
answer

280
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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. ...

4
votes

1
answer

264
views

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 ...

4
votes

0
answers

81
views

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 $L^\times$...

6
votes

0
answers

145
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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

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 ...

5
votes

0
answers

163
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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 $\...

2
votes

0
answers

99
views

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 ...

6
votes

3
answers

735
views

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 $\...

2
votes

0
answers

277
views

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. ...

1
vote

1
answer

129
views

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 $R$ ...

6
votes

1
answer

375
views

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 ...

3
votes

0
answers

163
views

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
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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 ...

0
votes

2
answers

444
views

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 ...

9
votes

0
answers

369
views

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 ...

6
votes

1
answer

531
views

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 ...

2
votes

0
answers

180
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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 $k$. ...

6
votes

0
answers

176
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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 $...

10
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0
answers

283
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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
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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 ...

10
votes

1
answer

432
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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(...

9
votes

1
answer

405
views

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'...

5
votes

0
answers

96
views

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 ...

1
vote

1
answer

114
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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^...

10
votes

1
answer

852
views

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 ...

10
votes

1
answer

1k
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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

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 ...

8
votes

1
answer

677
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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 \...

1
vote

2
answers

265
views

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

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
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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
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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 $D=\sum_i a_ip_i$ ...

5
votes

1
answer

325
views

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
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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 ...

4
votes

1
answer

438
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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 ...

0
votes

1
answer

601
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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 ...

9
votes

1
answer

3k
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Is there an explicit description of the maximal tamely ramified extension of $\mathbf Q_p$?

6
votes

3
answers

1k
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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
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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
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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
votes

2
answers

386
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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 ...

1
vote

1
answer

1k
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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 ...

11
votes

1
answer

1k
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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

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!

26
votes

3
answers

2k
views

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{...