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.
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$SL_2(\mathbb{Z}_p)$ extension of a local field
Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...
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Galois theory of ramified coverings vs classical Galois theory
That's an exact copy of my former MSE question I asked a couple of weeks ago and unfortunately not got the answer I was looking for.
The question adresses reuns' answer in this thread: Algebraic ...
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$0$-Dimensional $k$-varieties in the Grothendieck ring $K_0(V_k)$
Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. Let $K_0(V_k^0)$ be the Grothendieck ring of $0$-dimensional $k$-varieties. I also assume that $k$ is perfect.
(1) I ...
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Galois groups of special polynomials
This question is motivated by long experiments with GAP.
Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
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Galois group of polynomials related to Fibonacci and Catalan numbers
Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers.
Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$.
For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$.
And another ...
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Number of real roots of irreducible polynomials that are solvable by radicals
Let $n \geq 3$ be a natural number. Define the set $X_n$ as the set of natural numbers that appear as the number of real roots an irreducible polynomial of degree $n$ over $\mathbb{Q}$ which is ...
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Criterion for generic polynomials
Generic polynomials, which are recalled below, play an important role in the constructive aspects of the inverse Galois problem.
Definition. Let $P(\mathbf{t},X)$ be a monic polynomial in $\mathbb{Q}...
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Can completely multiplicative functions be extended to $\overline{\mathbb{Q}}$ or further?
I'm looking for a subject of study that handles the following question. I'm not the most familiar with algebra; I have a strong working knowledge and that's about it, but I've been considering ...
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Does "tensoring" with a fixed field preserve Galois extensions of finite fields?
Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}...
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The construction of the 257gon
If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...
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Grothendieck's Galois theory without finiteness hypotheses
This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any ...
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"sparsifying" a binary (over the field F2) matrix
Assume I have a matrix $A \in GF(2)$, i.e., $A_{i,j} \in \{0, 1\}$ and the sum is modulo 2. Is there any known algorithms/methods to sparsify (reduce the number of non-zero entries) $A$ while keeping ...
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Is there a Galois theory for deformations of curves?
I have some general questions about the deformations of Galois covers of curves. Suppose we are given a $G$-Galois cover $k[[z]]/k[[x]]$, where $k$ is algebraically closed of characteristic $p>0$. ...
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Expressing every algebraic number using roots of trinomials?
This question is a continuation of Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...
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Galois group for 0-dimensional motives
It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out.
One can ...
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Galois group of a product of polynomials
How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
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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 ...
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Why does $E\otimes_KE\cong EG$ imply that Galois theory works?
This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the Galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$.
"It is ...
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Given an abelian galois map of curves, what are the principal divisors on the source fixed by the galois group?
Let $f:X\rightarrow Y$ be an abelian galois map (not necessarily unramified) of nonsingular complete curves over algebraically closed $k$, where the order of the galois group $A$ is coprime to the ...
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Degree of prime power of an element [closed]
Let $p$ and $q$ be two distinct primes. For a field $F$, assume that $\deg(\alpha, F)=p$.
Is it necessarily true that $\deg(\alpha^q, F)=p$? Is there any counterexample?
It is not an exercise problem ...
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Properties of Mod $\ell^m$ Galois representation associated to modular form
(Sorry for my poor english..)
Let $F(z)\in S_{2k}(SL_2(\mathbb{Z})$) be a newform and $\ell$ be a prime larger than $3$. Let $K$ be a some number field and $v$ be a prime of $K$ over $\ell$. Let $K_v$...
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How to (easily) obtain the splitting field for dihedral extensions
Let $f(x) \in {\mathbb Q}[x]$ be a polynomial that is irreducible over ${\mathbb Q}$ with $D_{n}$ (the dihedral group of order $2n$) as its Galois group. Let $\alpha$ be a root of $f(x)$ and put $K={\...
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The inverse Galois problem and the Monster
I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, ...
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Cubic polynomials over finite fields whose roots are quadratic residues or non-residues
For a cubic polynomial $f(x)=x^3+x^2+\frac{1}{4}x+c$ over $\mathbb{F}_q$, where $q$ is a odd prime power, I find that for a lot of $q$, there does not exist $c\in\mathbb{F}_q$ such that $f$ has three ...
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English literature close to "Algébre et Théories Galoisiennes" by Régine and Adrien Douady
I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...
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Any way around Abel's impossibility theorem?
Abel's impossibility theorem states that the roots of a general polynomial (of degree 5 or higher) cannot be written using arithmetic operations and radicals. Radicals are solutions of a specific ...
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Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group
There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1):
$S$ is a Henselian trait (i.e. the spectrum of a henselian discrete ...
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Algorithm to compute minimal polynomials
Suppose $L/K$ is a finite Galois extension of fields of degree $n$. Suppose that we know an irreducible polynomial $f\in K[x]$ such that $L\cong K[x]/(f)$. Suppose also that we know the Galois group ...
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Non-linear Galois descent
This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...
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Lang's proof of the Shimura's exact sequence
Let $E$ be an elliptic curve having invariant $j$ and defined over $\mathbf C(j)$. Let $\sigma$ be an automorphism of $F_{\mathbf C}$, the field of all modular functions of all levels. Let $p$ be a ...
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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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split integral model of a reductive group
Let $F$ be a number field, $p\in\mathbb{Z}$ a prime which is unramified in $F$ and $G$ a connected reductive group over $F$. Moreover $G$ is supposed to be quasi-split over $p$.
Does there exist a ...
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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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Fontaine - Wintenberger field of norms and imperfect case
Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
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Local Class field theory for curves over $p$-adic fields
Let $X$ be a smooth curve over $\operatorname{Spec}\mathbb{Q}_p$ and $P\in X(\mathbb{Q}_p)$. Let $K_P\simeq \mathbb{Q}_p((T))$ denote the completion of the function field of $X$ at $P$. What is known ...
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p-adic field extension of degree 2n without a subfield of degree 2?
I need an example of a p-adic field extention $L/F$ of degree $[L:F]=2n$ without a subfield $K\subset L$ of degree $[K:F] = 2$.
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Aligned roots of irreducible polynomials
It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
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Collinear Galois conjugates
This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me.
Let $p$ be an irreducible polynomial with integer ...
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Linear independence of approximate square roots
From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
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Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
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Abel-Ruffini theorem for systems of polynomial equations
I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...
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Abel and Galois (and Arnold)
Question Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
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Galois group of constructible numbers
Let $\mathcal{C}$ be the field of constructible numbers, that is, the complex numbers constructible by compass and straightedge. It can be shown that it consists of all the numbers obtainable by ...
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Is there an algorithm to compute a Belyi map for the Riemann surface?
Let $y^2=x^5-x-1$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $\{0,...
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Galois theory and Lie theory [closed]
Galois theory talks about solving algebraic equations by radicals and the motivation for Lie to develop his theory is to study differential equations.
I want to understand what is happening in common ...
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Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group
In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...
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Name and properties of this combination of group algebra and semidirect product?
Given a field $k$, a group $G$, and a homomorphism $\phi : G \to \mathrm {Aut} (k)$, we can define a ring $\widehat {k [G]}_\phi$ as follows: As an abelian group it is isomorphic to the group algebra $...
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What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$?
This question began as Why are procyclic subgroups of Galois groups of number fields free profinite?, which fizzled out, but which garnered some helpful comments from YCor.
Let $K$ be a field, take $\...
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Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
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Galois action on Grothendieck ring of varieties
Let $k$ be a field and let $\overline{k}$ be its algebraic closure. Let $K_0(Var/\overline{k})$ be the Grothendieck ring of algebraic varieties over $\overline{k}$.
Is it true that the natural ...
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