When is the ring of integers of a number field coverable?

A commutative ring R is said to be coverable if it is the union of its proper subrings and said to be finitely coverable if it is the union of a finite number of them. In the latter case, we denote by {\sigma}(R) the minimal number of required subrings. In this paper, we give necessary and sufficient conditions for the ring of integers A of a given number field to be finitely coverable and a formula for {\sigma}(A) is given which holds when they are met. The conditions are expressed in terms of the existence of common index divisors and (or) common divisors of values of polynomials

Résolution de certaines équations diophantiennes et propriétés de certains polynômes

Dans les quatre premiers chapitres de cette thèse, nous abordons quelques équations diophantiennes et leurs solutions. On démontre que l'équation y 2 = px(Ax2 + 2) n'admet qu'un maximum de six solutions entières où p est nombre premier et A > 1 est entier impair ; on démontre que l'équation Resx P(x), x2 + sx + t = a n'admet qu'un nombre ni de solutions (s, t) pour P un polynôme xe et a un entier autre que zéro ; on résout l'équation Fn−Fm = y a lorsque y ∈ {6, 11, 12} et on trouve une borne pour les solutions de Fn + Fm = y a dans le cas général ; et on démontre que si un nombre su sant d'entiers x consécutifs existent tels que P(x) est sous la forme mq lorsque q ≥ 2 est diviseur de deg P, alors P = Rq pour un certain polynôme R, ce qui nous permet de déduire l'existence d'une in nité de solutions à y q = P(x) à partir d'un nombre ni de telles solutions dans certains cas. Dans les six derniers chapitres, nous abordons plusieurs sujets reliés à la décomposition d'objets algébriques. Parmi les résultats, on présente quelques conditions sous lesquelles un polynôme ne peut pas être exprimé comme une composition de deux polynômes de degré inférieur ; on présente une nouvelle démonstration du théorème Carltiz-Lutz sur les polynômes de permutations ; on étudie la possibilité d'exprimer un polynôme comme une somme composée ou un produit composé de deux autres polynômes de degré inférieur ; on trouve une borne pour un des plus petits nombres premiers qui se décompose dans un corps imaginaire quadratique donné ; et on étudie la possibilité de recouvrir un anneau avec ses sous-anneauxThe rst four chapters of this thesis address some Diophantine equations and their solutions. We prove that the equation y 2 = px(Ax2 + 2) has at most six integer solutions (x, y) for p a prime and A > 1 an odd integer; we prove that the equation Resx P(x), x2 + sx + t = a has only nitely many integer solutions (s, t) for a xed polynomial P and nonzero integer a; we completely solve the equation Fn − Fm = y a for y ∈ {6, 11, 12} and bound the solutions for Fn + Fm = y a in general; and we prove that the existence of su ciently many consecutive integers x such that P(x) is of the form mq for q ≥ 2 dividing deg P implies that Rq for some polynomial R, providing criteria for deducing the existence of in nitely many solutions to y q = P(x) from the existence of nitely many solutions in some cases. In the last six chapters, we address various algebraic decomposition related topics. Among other results, we provide criteria which guarantee a polynomial cannot be written as a composition of two polynomials of smaller degree; we provide a new proof of the Carlitz-Lutz theorem on permutation polynomials; we study the possibility of expressing a polynomial as the composed sum or composed multiplication of two polynomials of smaller degree; we bound from below some of the smallest primes which split in an imaginary quadratic eld; and we study the possibility of covering a ring with its subring

A Formalization of Dedekind Domains and Class Groups of Global Fields

Submitted to the conference Interactive Theorem Proving 2021 (Rome, Italy)International audienceDedekind domains and their class groups are notions in commutative algebra that are essential in algebraic number theory. We formalized these structures and several fundamental properties, including number theoretic finiteness results for class groups, in the Lean prover as part of the mathlib mathematical library. This paper describes the formalization process, noting the idioms we found useful in our development and mathlib's decentralized collaboration processes involved in this project

Rings of small rank over a Dedekind domain and their ideals

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite

Extensions of discrete valuations and their ramification theory

We study how a discrete valuation v on a field K can be extended to a valuation of a finite separable extension L of K. The ramification theory of extensions of discrete valuations to a finite separable extension is very well established whenever the residue class field extension is separable. This is the so called classical ramification theory. We investigate the classical ramification theory and also the ramification theory of extensions of discrete valuations with an inseparable residue class field extension. We show that some results from classical ramification theory, such as Hilbert's different formula can be modified to be true for extensions of valuations with inseparable residue class field extensions, whereas many other classical results fail to hold

On the behavior of test ideals under finite morphisms

We derive transformation rules for test ideals and $F$-singularities under an arbitrary finite surjective morphism $\pi : Y \to X$ of normal varieties in prime characteristic $p > 0$. The main technique is to relate homomorphisms $F_{*} O_{X} \to O_{X}$, such as Frobenius splittings, to homomorphisms $F_{*} O_{Y} \to O_{Y}$. In the simplest cases, these rules mirror transformation rules for multiplier ideals in characteristic zero. As a corollary, we deduce sufficient conditions which imply that trace is surjective, i.e. $Tr_{Y/X}(\pi_{*}O_{Y}) = O_{X}$.Comment: 33 pages. The appendix has been removed (it will appear in a different work). Minor changes and typos corrected throughout. To appear in the Journal of Algebraic Geometr