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
5
votes
0
answers
222
views
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 ...
- 4,418
2
votes
0
answers
150
views
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 ...
- 667
4
votes
1
answer
243
views
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 $L$ (...
- 421
3
votes
1
answer
384
views
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,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
- 12.6k
3
votes
1
answer
341
views
A similar relationship between the generic cubic and the Lehmer quintic?
I. Comparison
It doesn't seem to be well-known that the generic cubic (prominent in this MO post) for $C_3 = A_3$,
$$x^3-nx^2+(n-3)x+1 = 0$$
has the nice property that its roots $a,b,c$, if in correct ...
- 12.6k
7
votes
2
answers
439
views
A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics
I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
- 12.6k
3
votes
1
answer
250
views
On the refined minimal ramification problem for $p$-groups
Let $p$ be a prime. The minimal ramification problem is to ask whether or not every finite $p$-group $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly ...
- 667
2
votes
2
answers
381
views
On V. Arnold's trinities regarding PSL(2,5), PSL(2,7), and PSL(2,11)?
Given the Ramanujan theta function,
$$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 $...
- 12.6k
5
votes
1
answer
273
views
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 ...
- 2,907
6
votes
0
answers
147
views
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}} =...
- 12.6k
-2
votes
1
answer
192
views
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 ...
- 147
6
votes
1
answer
533
views
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 ...
3
votes
3
answers
368
views
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 ...
- 12.6k
0
votes
1
answer
243
views
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}...
- 667
5
votes
1
answer
289
views
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 ...
- 12.6k
1
vote
1
answer
305
views
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\...
- 833
2
votes
1
answer
182
views
Measure on the places of $\bar{\mathbb Q}$
Consider the set $S$ of all places of $\mathbb Q$ (i.e. the set of all absolute values up to equivalence). Then we can consider $S$ as a measure space with the counting measure $\mu$. Therefore $\mu(\{...
- 321
0
votes
0
answers
80
views
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 ...
- 309
2
votes
1
answer
159
views
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 ...
- 309
27
votes
1
answer
2k
views
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 ...
- 63.1k
2
votes
1
answer
307
views
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$?
- 65
2
votes
0
answers
146
views
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^...
- 930
3
votes
1
answer
213
views
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'...
- 2,689
7
votes
1
answer
890
views
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 ...
- 325
3
votes
0
answers
128
views
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}^{\...
- 2,197
5
votes
1
answer
438
views
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 ...
- 815
5
votes
1
answer
511
views
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$....
- 1,322
6
votes
0
answers
375
views
How to construct this non-geometric mod $p$ Galois representation?
Let $ K $ be a totally real cubic field. I am concerned with the following kind of Inverse Galois Problem:
Is there a prime $ p>5 $ and a continuous representation $ \bar{\rho}:G_{K}\to {\rm
GL}_{...
- 863
1
vote
0
answers
135
views
Reference request: arithmetical implications of an ambient Galois extension
This is a cross-post of a question I asked on StackExchange. See there for further details.
Let $L/K$ be a Galois extension of algebraic number fields of finite degree over $\mathbb{Q}$, with group $G$...
- 1,388
6
votes
0
answers
220
views
Can we deduce that both $\alpha$ and $\beta$ are algebraic over F if $F[\alpha,\beta]=F(\alpha,\beta)$
So after reading David Cox. book on Galois Theory, it can be shown that “Given a field F, $F[\alpha]=F(\alpha)$ if and only if $\alpha$ is algebraic over F” (See Prop. 4.1.14).
Later, it was mentioned ...
- 325
-2
votes
1
answer
504
views
In Galois theory, why solvable groups must have their quotient groups be Abelian? [closed]
The definition of solvable groups can be regarded as two constraints, one is that there must be a sequence of normal subgroups, and the other is that the quotient groups between these sequences are ...
- 1
0
votes
0
answers
120
views
Colimits in the category of suplattices
I want to compute coequalizers in the category $\mathcal{S}up$ of complete lattices and $\bigvee$-preserving maps. One way (I think?) is to use the dual equivalence
$$
\mathcal{Sup} \leftrightarrows \...
- 147
8
votes
1
answer
355
views
The distribution of certain Galois groups
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
- 7,193
6
votes
2
answers
329
views
Algebraic numbers which prescribed degree which does not belong to some fields
In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
- 515
4
votes
0
answers
83
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 $p \nmid n$, $m$. ...
- 1,833
3
votes
1
answer
393
views
Completion of infinite degree extension of perfectoid fields is perfectoid?
Is completion of infinite degree extension of perfectoid fields perfectoid ?
It is known that finite extension of perfectoid fields is also perftoid from tilting correspondence, but what about ...
- 1,531
5
votes
2
answers
1k
views
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 ...
- 4,547
2
votes
0
answers
148
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 $\mathbb{C}$, ...
- 4,547
4
votes
0
answers
245
views
Dessins d'enfants and the absolute Galois group
If I am not mistaken, Alexander Grothendieck introduced dessins d'enfants as they are known today, in order to better understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}...
- 4,547
0
votes
0
answers
78
views
Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$
Let $K$ be a local field of positive characteristic.
I'm looking for a $K$ which satisfies the following condition.
Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n)...
- 1,531
4
votes
1
answer
354
views
Can a general quintic be solved using inverse beta regularized function?
Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
- 10.1k
5
votes
1
answer
345
views
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 ...
- 695
5
votes
1
answer
286
views
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 ...
- 135
7
votes
0
answers
149
views
Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
- 299
6
votes
2
answers
280
views
Cancellation of irreducibility for Galois conjugates
Motivation: Take an algebraic number $\lambda$. In my research, I've stumbled upon the question in which cases the expression $\sum_{\sigma \in S} \sigma(\lambda)$, where $S$ is a subset of the field ...
- 135
9
votes
2
answers
698
views
Non-trivial automorphisms and descent
In this expository paper by Low it says:
Roughly
speaking, a topos in the sense of Grothendieck is the category of sheaves on a
kind of generalised space whose “points” may have non-trivial ...
- 91
5
votes
1
answer
417
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 ...
- 15.6k
6
votes
2
answers
817
views
What is $\mathbb Q^{\mathrm{hypoab}}$?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}\newcommand{\ab}{\mathrm{ab}}$Let $G(\mathbb Q) = \Gal(\overline{\mathbb Q} / \mathbb Q)$ be the absolute Galois group. It's well-known that ...
- 63.9k
3
votes
1
answer
343
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. ...
- 15.6k
6
votes
1
answer
466
views
Is there a conjectured dependence on $n$ in van der Waerden's conjecture?
Bhargava 2021 proves van der Waerden's conjecture about Galois groups of random integer polynomials: over all $x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ with $a_k \in \{-H, \ldots, H\}$, the number of ...
- 1,798