We extend the notion of finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of centrally finite division rings. We first show that solving a finite ...embedding problem over such a division ring
H
is equivalent to finding a solution to some finite embedding problem over the center of
H
involving a polynomial constraint. We then show that every constant finite split embedding problem over the skew field of rational fractions
H
(
t
) with central indeterminate
t
has a solution, if the center of
H
is an ample field, thus providing with a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over skew fields of rational fractions
H
(
t, σ
), where
σ
denotes an arbitrary automorphism of
H
with finite order. Our results generalize previous works on the inverse Galois problem over skew fields.
Twisted skew G-codes Behajaina, Angelot; Borello, Martino; Cruz, Javier de la ...
Designs, codes, and cryptography,
06/2024, Volume:
92, Issue:
6
Journal Article
Peer reviewed
In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are in most cases algebras over a finite field, allow us to retrieve many of the well-known ...codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes.
Dans cet article, nous montrons que si H est un corps gauche de centre k et σ un automorphisme d'ordre fini de H tels que le sous-corps k〈σ〉 de k fixé par σ contient un corps ample, alors le problème ...inverse de Galois admet une réponse positive sur le corps H(t,σ) des fractions rationnelles tordus. De plus, si k〈σ〉 contient un corps qui est, soit réel clos, soit hensélien de caractéristique résiduelle nulle et contenant toutes les racines de l'unité, alors le groupe prolibre de rang dénombrable Fˆω est groupe de Galois sur H(t,σ).
In this article, we prove that if H is a skew field of center k and σ an automorphism of finite order of H such that the fixed subfield k〈σ〉 of k under the action of σ contains an ample field, then the inverse Galois problem has a positive answer over the skew field H(t,σ) of twisted rational fractions. Moreover, if k〈σ〉 contains either a real closed field, or an Henselian field of residue characteristic 0 and containing all roots of unity, then the profree group of countable rank Fˆω is a Galois group over H(t,σ).
Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and ...unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.
We study the analogue of the Collatz map in the polynomial ring Fpx, for any prime number p, and the corresponding dynamical system. We show that every f∈Fpx is eventually periodic in this system, in ...a quadratic number of iterations in deg(f), and describe explicitly all corresponding cycles. This extends a result of Hicks, Mullen, Yucas and Zavislak, who studied the case p=2. We also study the Collatz map in the formal power series ring Fpx, observe that in Fpx all but countably many power series generate divergent trajectories via iterations of this map, and characterize those power series that are eventually periodic.
Dans cet article, nous construisons, pour certains 2-groupes G, des extensions galoisiennes explicites E/ℚ(T) de groupe G vérifiant
E
∩
ℚ
¯
=
ℚ
. Nous fournissons aussi des progressions arithmétiques ...explicites d’entiers t₀ telles que la spécialisation
E
t
0
/
=
ℚ
de E/ℚ(T) en t₀ soit de groupe G.
In this paper, we construct, for some 2-groups G, explicit Galois extensions E/ℚ(T) of group G with
E
∩
ℚ
¯
=
ℚ
. We also provide explicit arithmetic progressions of integers t₀ such that the specialization
E
t
0
/
=
ℚ
of E/ℚ(T) at t₀ has Galois group G.
PDF
