Finite groups of units of finite characteristic rings

In \cite[Problem 72]{Fuchs60} Fuchs asked the following question: which groups can be the group of units of a commutative ring? In the following years, some partial answers have been given to this question in particular cases. The aim of the present paper is to address Fuchs' question when $A$ is a {\it finite characteristic ring}. The result is a pretty good description of the groups which can occur as group of units in this case, equipped with examples showing that there are obstacles to a "short" complete classification. As a byproduct, we are able to classify all possible cardinalities of the group of units of a finite characteristic ring, so to answer Ditor's question \cite{ditor}

On wild extensions of a p-adic field

In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the isomorphism classes of these extensions and some Kummer extensions of a suitable field F containing K. We then describe such classes in terms of the representations of Gal(F/K). Finally, for k = 2 and for each possible Galois group G, we count the number of isomorphism classes of the extensions whose normal closure has a Galois group isomorphic to G. As a byproduct, we get the total number of isomorphism classes

Local-global questions for divisibility in commutative algebraic groups

This is a survey focusing on the Hasse principle for divisibility of pointsin commutative algebraic groups and its relation with the Hasse principle fordivisibility of elements of the Tate-Shavarevich group in the Weil-Ch\^{a}teletgroup. The two local-global subjects arose as a generalization of someclassical questions considered respectively by Hasse and Cassels. We describethe deep connection between the two problems and give an overview of thelong-established results and the ones achieved during the last twenty years,when the questions were taken up again in a more general setting. Inparticular, by connecting various results about the two problems, we describehow some recent developments in the first of the two local-global questionsimply an answer to Cassel's question, which improves all the results publishedbefore about that problem. This answer is best possible over $\mathbb{Q}$. Wealso describe some links with other similar questions, as for examples theSupport Problem and the local-global principle for existence of isogenies ofprime degree in elliptic curves.<br

Diversity in Parametric Families of Number Fields

Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/\log N distinct, where c>0. We prove that there are at least N/(\log N)^{1-c} distinct fields, where c>0.Comment: Minor inaccuracies detected by the referees are correcte

Composite factors of binomials and linear systems in roots of unity

In this paper we completely classify binomials in one variable which have a nontrivial factor which is composite, i.e., of the shape g(h(x)) for polynomials g, h both of degree &gt; 1. In particular, we prove that, if a binomial has such a composite factor, then deg g 64 2 (under natural necessary conditions). This is best-possible and improves on a previous bound deg g 64 24. This result provides evidence toward a conjecture predicting a similar bound when binomials are replaced by polynomials with any given number of terms. As an auxiliary result, which could have other applications, we completely classify the solutions in roots of unity of certain systems of linear equations

On certain infinite extensions of the rationals with Northcott property

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that \IQ_{ab}^{(d)}, the maximal abelian subfield of the field generated by all algebraic numbers of degree at most $d$, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. \IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...) has the Northcott property if and only if $2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$ tends to infinity

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)