Galois towers over non-prime finite fields

Alp Bassa; Peter Beelen; Arnaldo Garcia; Henning Stichtenoth

Displaying similar documents to “Galois towers over non-prime finite fields”

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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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...

Random Galois extensions of Hilbertian fields

Lior Bary-Soroker, Arno Fehm (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $L$ be a Galois extension of a countable Hilbertian field $K$ . Although $L$ need not be Hilbertian, we prove that an abundance of large Galois subextensions of $L / K$ are.

Counting discriminants of number fields

Henri Cohen, Francisco Diaz y Diaz, Michel Olivier (2006)

Journal de Théorie des Nombres de Bordeaux

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For each transitive permutation group $G$ on $n$ letters with $n \leq 4$ , we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $ℚ$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$ .

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

Magali Bouffet (2003)

Bulletin de la Société Mathématique de France

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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 $ℂ ((z))$ .

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

James E. Carter (2007)

Colloquium Mathematicae

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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...

An equicharacteristic analogue of Hesselholt's conjecture on cohomology of Witt vectors

Amit Hogadi, Supriya Pisolkar (2013)

Acta Arithmetica

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Let L/K be a finite Galois extension of complete discrete valued fields of characteristic p. Assume that the induced residue field extension $k_{L} / k_{K}$ is separable. For an integer n ≥ 0, let $W_{n} (_{L})$ denote the ring of Witt vectors of length n with coefficients in $_{L}$ . We show that the proabelian group ${H^{1} (G, W_{n} (_{L}))}_{n \in ℕ}$ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture, which was proved before when the discrete valued fields are of mixed characteristic.

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

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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 $D_{16}$ .

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields $K$ having isomorphic absolute Abelian Galois groups $A_{K}$ , we study any such issue for arbitrary number fields $K$ . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some $p$ -adic obstructions coming from the global units of $K$ . By restriction to the $p$ -Sylow subgroups of $A_{K}$ and assuming the Leopoldt conjecture we...

On the subfields of cyclotomic function fields

Zhengjun Zhao, Xia Wu (2013)

Czechoslovak Mathematical Journal

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Let $K = 𝔽_{q} (T)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f (T) \in 𝔽_{q} [T]$ with positive degree, denote by $Λ_{f}$ the torsion points of the Carlitz module for the polynomial ring $𝔽_{q} [T]$ . In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K (Λ_{P})$ of degree $k$ over $𝔽_{q} (T)$ , where $P \in 𝔽_{q} [T]$ is an irreducible polynomial of positive degree and $k > 1$ is a positive divisor of $q - 1$ . A formula for the analytic class...

Greatest common divisors of $u - 1$ , $v - 1$ in positive characteristic and rational points on curves over finite fields

Pietro Corvaja, Umberto Zannier (2013)

Journal of the European Mathematical Society

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In our previous work we proved a bound for the $g c d (u - 1, v - 1)$ , for $S$ -units $u, v$ of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in [3]. As pointed out by Silverman, the exact analogue does not work for function fields in positive characteristic. In the present work, we investigate possible extensions in that direction; it turns out that under suitable assumptions some of the results still hold. For instance we prove Theorems 2 and 3...

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

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We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form $H = 1 / 2 \sum_{i = 1}^{n} p ²_{i} + V (q)$ , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form $V = V_{k} + V_{K}$ , where $V_{k}$ and $V_{K}$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples...