The regular inverse Galois problem over non-large fields

Jochen Koenigsmann

Displaying similar documents to “The regular inverse Galois problem over non-large fields”

Galois towers over non-prime finite fields

Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth (2014)

Acta Arithmetica

Similarity:

We construct Galois towers with good asymptotic properties over any non-prime finite field $_{ℓ}$ ; that is, we construct sequences of function fields = (N₁ ⊂ N₂ ⊂ ⋯) over $_{ℓ}$ of increasing genus, such that all the extensions $N_{i} / N_{1}$ are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with...

The Sylow p-Subgroups of Tame Kernels in Dihedral Extensions of Number Fields

Qianqian Cui, Haiyan Zhou (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let F/E be a Galois extension of number fields with Galois group $D_{2 ⁿ}$ . In this paper, we give some expressions for the order of the Sylow p-subgroups of tame kernels of F and some of its subfields containing E, where p is an odd prime. As applications, we give some results about the order of the Sylow p-subgroups when F/E is a Galois extension of number fields with Galois group $D_{16}$ .

Some remarks on Hilbert-Speiser and Leopoldt fields of given type

James E. Carter (2007)

Colloquium Mathematicae

Similarity:

Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers $O_{L}$ in L is free as a module over the associated order $_{L / K}$ . We also give examples, some of which show that this result can still hold without the assumption that...

Polynomials over $Q$ solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

Similarity:

The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_{n}$ , can be embedded in any central extension of $A_{n}$ if and only if $n \equiv 0 (m o d 8)$ , or $n \equiv 2 (m o d 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$ , every central extension of $A_{n}$ occurs as a Galois group over $Q$ .

Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett, Vandita Patel, Samir Siksek (2016)

Acta Arithmetica

Similarity:

Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization ${(x - 1)}^{k} + x^{k} + {(x + 1)}^{k} = z^{n}$ (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The...

Differential Galois Theory for an Exponential Extension of $ℂ ((z))$

Magali Bouffet (2003)

Bulletin de la Société Mathématique de France

Similarity:

In this paper we study the formal differential Galois group of linear differential equations with coefficients in an extension of $ℂ ((z))$ by an exponential of integral. We use results of factorization of differential operators with coefficients in such a field to give explicit generators of the Galois group. We show that we have very similar results to the case of $ℂ ((z))$ .

On the generalized Fermat equation over totally real fields

Heline Deconinck (2016)

Acta Arithmetica

Similarity:

In a recent paper, Freitas and Siksek proved an asymptotic version of Fermat’s Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A x^{p} + B y^{p} + C z^{p} = 0$ , where A, B, C are odd integers belonging to a totally real field.

On a notion of “Galois closure” for extensions of rings

Manjul Bhargava, Matthew Satriano (2014)

Journal of the European Mathematical Society

Similarity:

We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an $S_{n}$ degree $n$ extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

Similarity:

In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

Galois co-descent for étale wild kernels and capitulation

Manfred Kolster, Abbas Movahhedi (2000)

Annales de l'institut Fourier

Similarity:

Let $F$ be a number field with ring of integers $o_{F}$ . For a fixed prime number $p$ and $i \geq 2$ the étale wild kernels $W K_{2 i - 2}^{\overset{´}{e} t} (F)$ are defined as kernels of certain localization maps on the $i$ -fold twist of the $p$ -adic étale cohomology groups of $spec o_{F} [\frac{1}{p}]$ . These groups are finite and coincide for $i = 2$ with the $p$ -part of the classical wild kernel $W K_{2} (F)$ . They play a role similar to the $p$ -part of the $p$ -class group of $F$ . For class groups, Galois co-descent in a cyclic extension $L / F$ is described by the ambiguous class formula given...

The Galois group of $X^{p} + a X^{s} + a$

B. Bensebaa, A. Movahhedi, A. Salinier (2008)

Acta Arithmetica

Similarity:

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

Similarity:

Let $L / K$ be an extension of algebraic number fields, where $L$ is abelian over $ℚ$ . In this paper we give an explicit description of the associated order $𝒜_{L / K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_{L}$ , the ring of integers of $L$ , is then isomorphic to $𝒜_{L / K}$ . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that $𝒜_{L / K}$ is the maximal order if $L / K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to $ℚ^{(m^{'})} / K$ , where $m^{'}$ is the conductor...

Supremum properties of Galois-type connections

Árpád Száz (2006)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In a former paper, motivated by a recent theory of relators (families of relations), we have investigated increasingly regular and normal functions of one preordered set into another instead of Galois connections and residuated mappings of partially ordered sets. A function $f$ of one preordered set $X$ into another $Y$ has been called (1) increasingly $g$ -normal, for some function $g$ of $Y$ into $X$ , if for any $x \in X$ and $y \in Y$ we have $f (x) \leq y$ if and only if $x \leq g (y)$ ; (2) increasingly $ϕ$ -regular, for some function $ϕ$ ...