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 the integral values of a curious recurrence

We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of diophantine equations

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

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

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

On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,\zeta_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $\zeta_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(\zeta_{p^r},\sqrt[m_1]{a})$, with $a\in K(\zeta_{p^r})$ and $m_1|p^r$. In addition, when $K$ is a number field, we produce an upper bound to the logarithmic height of the discriminant of the extension $K(\mathcal{E}[m])/K$, for all $m\geq 3$. As a consequence, we give an explicit effective version of the hypotheses of the local-global divisibility problem in elliptic curves over number fields

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

An equivalence between local fields

AbstractThe p-component of the index of a number field K depends only on the completions of K at the primes over p. In this paper we define an equivalence relation between m-tuples of local fields such that, if two number fields K and K′ have equivalent m-tuples of completions at the primes over p, then they have the same p-component of the index. This equivalence can be interpreted in terms of the decomposition groups of the primes over p of the normal closures of K and K′

On Fuchs' Problem about the group of units of a ring

In cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a previous paper cite{DDcharp} we dealt with finite characteristic rings. In this paper we consider Fuchs' question for finite groups and we address this problem in two cases. Firstly, we study the case of torson-free rings and we obtain a complete classification of the finite groups of units which arise in this case. Secondly, we examine the case of characteristic zero rings obtaining, a pretty good description of the possible groups of units equipped with families examples of both realizable and non-realizable groups. The main tools to deal with this general case are the Pearson and Schneider splitting of a ring cite{PearsonSchneider70}, our previous results on finite characteristic rings cite{DDcharp} and our classification of the groups of units of torsion-free rings. As a consequence of our results we completely answer Ditor's question cite{ditor} on the possible cardinalities of the group of units of a ring