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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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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why is $R := \mathbb{Z}p +XF[[X]]$ an almost valuation domain?
Recall that an integral domain $R$ with quotient field $K$ is
an almost valuation domain if for every $0 \not= x \in K$, there is a positive integer
$n$ (depending on $x$) such that $x^n \in R$ or $x^{...
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Surjectivity of a norm map over $ \mathbb{Q} $
Suppose $ (L/L') $ is an galois extension , where both fields are extension of $Q$ of $\dim n $ and $\dim n^{'}$ respectively.Suppose we consider the norm map $ Nr_{(L/L')} :L \rightarrow L' $. ...
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Is there a name for objects all of whose endomorphisms are automorphisms?
I am looking for a descriptive adjective to describe the following special property that some objects in some categories enjoy: their endomorphism monoids are groups. Of course, one way this could ...
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An extension of the Galois theory of Grothendieck
This question is about Joyal and Tierney's famous An extension of the Galois theory of Grothendieck. One of the main results states (see the MathSciNet review by Peter Johnstone):
Joyal and Tierney's ...
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Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
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Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
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Irreducible components of a cyclic extension over $ \mathbb{Q} $
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then ...
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Do surface groups embed into PSL_2 over a real quadratic integer ring?
$\DeclareMathOperator\PSL{PSL}$ Let $ \mathbb{Z} $ be the ring of integers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma_g $ be a surface of genus $ g \geq 2 $. Let $ \pi_1(\Sigma_g) $ be ...
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Structure of coefficients of polynomials giving a specified Galois group
For any $a = [a_0; \dots; a_n]\in \mathbb{P}^n(\mathbb{Q})$, the corresponding Galois group $G_a$ of $f(X) = a_n X^n + \cdots + a_1 X + a_0\in \mathbb{Q}[X]$ is a subgroup of $S_n$. (I'm interested ...
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The upper bounds on rank $ 2 $ real matrices
Let $ A_{n}(F) $ be the collection of all skew-symmetric matrices over the field $ F $ ($\operatorname{char} F \neq 2 $). Let M be a subspace of $ A_{n}(F) $ such that all non zero elements have rank ...
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Simple proof of the equivalence between two definitions of étale
This question shouldn't be too hard to answer, but I'm looking for the most streamlined approach.
Let $K$ be a field and let $L$ be a finite dimensional field extension of $K$. I am interested in two ...
CommunityBot
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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
$$
u^2+v^2=z\overline{z}=\sqrt{a^2+b^2}
$$
...
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Restricting maps between strict henselisations
$\require{AMScd}$I am currently thinking about (strict) henselisations but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict ...
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What do dessins tell us about the absolute Galois group?
I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single ...
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If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?
I posed this question on Math.Stackexchange (see here) but until now there was no response. This made me decide to give it a try here.
Let $k\subseteq F$ denote an algebraic field extension and let $\...
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Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field
A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
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What are "nearly initial" objects really called?
Definition. Call an object $X$ of a category $\mathbf{C}$ nearly initial iff firstly, it is weakly initial, and secondly, for all objects $Y$ and all morphisms $f,g : X \rightarrow Y$, there exists an ...
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Problem concerning about an $n$-subspace of $ A_{n}(F) $
Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...
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Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's Esquisse he claims that the action of
$$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\...
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Existence of an irreducible polynomial that does not divide $x^n + a$
The question:
Can one characterize all fields $K$ over which there exists an irreducible polynomial $f(x)$ that does not divide a polynomial of the form $x^n + a$?
Examples:
Such a polynomial clearly ...
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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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Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $6$
Let us consider a primitive $7^{\text{th}}$ root of unity $\eta$. Then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6}$. So the dimension of the $\mathbb{Q}$-...
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Resolvent is minimal polynomial for universal splitting algebra
Given a degree $n$ monic $f\in A[x]$ write $\mathrm{Split}_Af$ for its universal splitting algebra, constructed by taking the quotient of $A[x_1,\dots ,x_n]$ by the Vieta formulas. This is the initial ...
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When is the extension $L(S)/L$ Galois and totally ramified?
Let $L$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ with uniformizer $\pi$. Let us consider the polynomial ring $L[x_1,x_2,\dotsc,x_l]$ in $l$-...
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Simultaneous Galois closure
For a finite separable extension $L/K$ of fields, there exists a Galois closure, which is a finite field extension $\tilde L/L/K$ where $\tilde L/K$ is Galois. (given by the compositum of $\sigma L$, ...
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Is there a measurable isomorphism ${\mathbb C}\to {\mathbb C}_p$?
Let $p$ be a prime and ${\mathbb C}_p$ be the completion of the algebraic closure $\overline{{\mathbb Q}_p}$. This field is isomorphic to $\mathbb C$. Both fields come with natural absolute values but ...
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Blow-ups of surfaces over a field
Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure.
Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
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Maximal common isotropic subspace for a finite family of skewforms
Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
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Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible
Let $A_{n}(\mathbb{Q}) $ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q}) $.
If all the non-zero matrices in $N$ are ...
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How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]
Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$.
Let $G$ be the group ...
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Computing the Galois group of a polynomial
Does there exist an algorithm which computes the Galois group of a polynomial
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of $p(...
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Field extension generated by the roots of multivariate-polynomials
Let us consider a field $K$ of characteristic $0$. Then we know that any finite extension $L$ of $K$, which is a Galois extension as well, is produced the roots of a separable polynomial $f(x) \in K[x]...
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Does any cubic polynomial become reducible through composition with some quadratic?
What I mean to ask is this:
given an irreducible cubic polynomial $P(X)\in \mathbb{Z}[X]$ is there always a quadratic $Q(X)\in \mathbb{Z}[X]$ such that $P(Q)$ is reducible (as a polynomial, and then ...
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On the characteristic polynomial of the Vandermonde matrix
Let $A_n$ be the $n \times n$-Vandermonde matrix (see for example https://en.wikipedia.org/wiki/Vandermonde_matrix )viewed as a matrix over the fraction field of the polynomial ring over a field $K$ (...
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Nef and pseudo-effective divisors over non algebraically closed fields
Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
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Inverse Galois problem on simple groups
Im trying to find a way to connect a possible solution of the inverse Galois problem on simple groups to a more general solution on any finite group.
I've tryied to mess with the embedding problem for ...
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Galois module theory: from global to local
Let $L/\mathbb{Q}$ be a finite Galois extension with Galois group $G$. It is well known that the ring of integers $\mathcal{O}_L$ is free over its associated order $\mathfrak{A}_{L/\mathbb{Q}}=\{x\in \...
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Original proof of Chebotarev's density theorem
As the title suggests, I am currently trying to understand Chebotarev's original proof of his density theorem, based on the proof in the appendix here. I am fully on-board with the cyclotomic ...
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References for Hopf Galois module theory
I am a first-year PhD student and I am really interested in Galois module theory, both in a "classical" and in a "nonclassical" sense. In the last months I have been reading about ...
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Simplification of links between idele class group and étale cohomology
I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
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The Galois resolvent in Lagrange
In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). ...
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Minimum product of degrees of generators of finite field extension
Suppose $L/K$ is a finite extension of fields. When is it true that $$\min_{\substack{\{\alpha_1, \ldots, \alpha_n\} \\ L = K(\alpha_1, \ldots, \alpha_n)}} \left(\prod_{i=1}^n [K(\alpha_i): K] \right) ...
CommunityBot
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$p$-adic Hodge theoretic properties of global Galois representations via $\ell$-Frobenii
Let $G_{\mathbb{Q},S} = \mathrm{Gal}(\mathbb{Q}_S/\mathbb Q)$ where $\mathbb Q_S$ is the largest algebraic extension of $\mathbb Q$ unramified outside a finite set of places $S$. Then the union over $\...
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Reducibility of a cubic over a number field
Given an extension $K = \mathbb{Q}(\alpha)$ by a cubic polynomial $g(x)\in \mathbb{Q}[x]$ (not necessarily Galois extension) is there a criterion for a cubic polynomial $f(x) \in K[x]$ to be reducible ...
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Centralizer of the absolute Galois group of a number field
By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
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A quantity associated to a field extension
Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space.
A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...
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Can every square root be represented as a linear combination on roots of unity? [duplicate]
Messing around, I noticed that
$$\sqrt{2}=e^{i\pi /4}+e^{-i\pi /4}$$
$$\sqrt{3}=e^{i\pi /6}+e^{-i\pi /6}$$
and (even more surprisingly)
$$\sqrt{5}=e^{2\pi i/5}-e^{4\pi i /5}-e^{6\pi i /5}+e^{8 \pi i /...
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An explicit isomorphism $K^\times/K^{\times p} \cong K^\flat/\wp K^\flat$ where $K \supset \mu_p$ is perfectoid field of mixed characteristic $(0, p)$
Let $K$ be a perfectoid field of mixed characteristic $(0, p)$, i.e. $K$ has characteristic $0$ but its residue field has characteristic $p$. Further suppose that $K$ contains all the $p$th roots of ...
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Effective Hilbert's Irreducibility Theorem and Irreducibility of $f(x)+1$, $f(x)\in\mathbb{Q}[x]$ reducible
Take a reducible polynomial $f(x)\in\mathbb{Q}[x]$. I am interested in the question: is $f(x)+1$ irreducible over $\mathbb{Q}$?
For $f (x) = (x −a_1) · · · (x −a_m)$ with distinct $a_1,\ldots, a_m\in \...
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