Galois groups as quotients of Polish groups

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an [Formula: see text] normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over [Formula: see text]. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932] which says that for any strong type defined on a single complete type over [Formula: see text], smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups.

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On the Galois group of three classes of trinomials

AIMS Mathematics ◽

10.3934/math.2022013 ◽

2021 ◽

Vol 7 (1) ◽

pp. 212-224

Author(s):

Lingfeng Ao ◽

◽

Shuanglin Fei ◽

Shaofang Hong

Keyword(s):

Symmetric Group ◽

Galois Group ◽

Prime Number ◽

Galois Groups

<abstract>Let $ n\ge 8 $ be an integer and let $ p $ be a prime number satisfying $ \frac{n}{2} <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ T_{n, p, k}(x): = x^n+n^kp^{(n-1-p)k}x^p+n^kp^{nk}, $\end{document} </tex-math></disp-formula> <disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ S_{n, p}(x): = x^n+p^{n(n-1-p)}n^px^p+n^pp^{n^2} $\end{document} </tex-math></disp-formula> and <disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ E_{n, p}(x): = x^n+pnx^{n-p}+pn^2 $\end{document} </tex-math></disp-formula> are the full symmetric group $ S_n $ under several conditions. This extends the Cohen-Movahhedi-Salinier theorem on the irreducible trinomials $ f(x) = x^n+ax^s+b $ with integral coefficients.</abstract>

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Calculating Galois groups of third-order linear differential equations with parameters

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500389 ◽

2018 ◽

Vol 20 (04) ◽

pp. 1750038

Author(s):

Andrei Minchenko ◽

Alexey Ovchinnikov

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Galois Group ◽

Galois Groups ◽

Linear Differential Equations ◽

Unipotent Radical ◽

Third Order ◽

Differential Galois Group ◽

Bessel Differential Equation ◽

Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

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Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-045-x ◽

2012 ◽

Vol 55 (1) ◽

pp. 38-47

Author(s):

William Butske

Keyword(s):

Galois Group ◽

Galois Groups ◽

Two Dimensional ◽

Finite Algebras ◽

Genus 2

AbstractZarhin proves that if C is the curve y2 = f (x) where Galℚ(f(x)) = Sn or An, then . In seeking to examine his result in the genus g = 2 case supposing other Galois groups, we calculate for a genus 2 curve where f (x) is irreducible. In particular, we show that unless the Galois group is S5 or A5, the Galois group does not determine .

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On Galois groups of class two extensions over the rational number field

Nagoya Mathematical Journal ◽

10.1017/s0027763000024867 ◽

1979 ◽

Vol 75 ◽

pp. 121-131 ◽

Cited By ~ 2

Author(s):

Susumu Shirai

Keyword(s):

Nilpotent Group ◽

Galois Group ◽

Number Field ◽

Direct Product ◽

Rational Number ◽

Nilpotency Class ◽

Abelian Extension ◽

Galois Groups ◽

Classification Theory ◽

Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

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Computing the Galois group of a polynomial over a p-adic field

International Journal of Number Theory ◽

10.1142/s179304212050092x ◽

2020 ◽

Vol 16 (08) ◽

pp. 1767-1801 ◽

Cited By ~ 1

Author(s):

Christopher Doris

Keyword(s):

Galois Group ◽

Galois Groups ◽

Naive Algorithm

We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of [Formula: see text] of degrees 18, 20 and 22, tables of which are available online.

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GALOIS GROUPS AND AN OBSTRUCTION TO PRINCIPAL GRAPHS OF SUBFACTORS

International Journal of Mathematics ◽

10.1142/s0129167x07003996 ◽

2007 ◽

Vol 18 (02) ◽

pp. 191-202 ◽

Cited By ~ 8

Author(s):

MARTA ASAEDA

Keyword(s):

On 2-Groups as Galois Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-064-3 ◽

1995 ◽

Vol 47 (6) ◽

pp. 1253-1273 ◽

Cited By ~ 14

Author(s):

Arne Ledet

Keyword(s):

Galois Group ◽

Direct Product ◽

Galois Extension ◽

Quaternion Algebra ◽

Abelian Groups ◽

Galois Groups ◽

Embedding Problem ◽

Brauer Group ◽

Characteristic 2 ◽

Cyclic Kernel

AbstractLet L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

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Classifying Galois groups of small iterates via rational points

International Journal of Number Theory ◽

10.1142/s1793042118500872 ◽

2018 ◽

Vol 14 (05) ◽

pp. 1403-1426

Author(s):

Wade Hindes

Keyword(s):

Galois Group ◽

Hyperelliptic Curve ◽

Rational Points ◽

Galois Groups ◽

Quadratic Polynomials

We establish several surjectivity theorems regarding the Galois groups of small iterates of [Formula: see text] for [Formula: see text]. To do this, we use explicit techniques from the theory of rational points on curves, including the method of Chabauty–Coleman and the Mordell–Weil sieve. For example, we succeed in finding all rational points on a hyperelliptic curve of genus 7, with rank 5 Jacobian, whose points parametrize quadratic polynomials with a “newly small” Galois group at the fifth stage of iteration.

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Independence of ℓ \ell -adic representations of geometric Galois groups

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0024 ◽

2018 ◽

Vol 2018 (736) ◽

pp. 69-93 ◽

Cited By ~ 1

Author(s):

Gebhard Böckle ◽

Wojciech Gajda ◽

Sebastian Petersen

Keyword(s):

Galois Group ◽

Finite Type ◽

Algebraic Closure ◽

Finite Extension ◽

Field Extension ◽

Galois Groups ◽

Closed Field ◽

Absolute Galois Group ◽

Arbitrary Characteristic ◽

Cohomology Modules

AbstractLetkbe an algebraically closed field of arbitrary characteristic, let{K/k}be a finitely generated field extension and letXbe a separated scheme of finite type overK. For each prime{\ell}, the absolute Galois group ofKacts on the{\ell}-adic étale cohomology modules ofX. We prove that this family of representations varying over{\ell}is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure ofKof the kernels of the representations for all{\ell}become linearly disjoint over a finite extension ofK. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

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Computation of Galois groups of rational polynomials

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157013000302 ◽

2014 ◽

Vol 17 (1) ◽

pp. 141-158 ◽

Cited By ~ 10

Author(s):

Claus Fieker ◽

Jürgen Klüners

Keyword(s):

Galois Group ◽

Galois Theory ◽

Large Degree ◽

Rational Numbers ◽

Galois Groups ◽

Multivariate Polynomial ◽

Theoretical Methods ◽

Algorithmic Solution ◽

Rational Polynomials ◽

General Method

AbstractComputational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar’s method. Computationally, one of the key challenges in the application of Stauduhar’s method is to find, for a given pair of groups $H<G$, a $G$-relative $H$-invariant, that is a multivariate polynomial $F$ that is $H$-invariant, but not $G$-invariant. While generic, theoretical methods are known to find such $F$, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.

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