A Short Review on p-Adic Numbers (original) (raw)

P-Adic metric preserving functions and their analogues

Mathematica Slovaca

The p-adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved.

ON THE ADDITIVE STRUCTURE OF Qp AND THE NATURE OF p-ADIC POWER SERIES

2011

This paper focuses on describing the nature of the field Qp, and understanding properties of p-adic power series. We begin by defining a norm |·|p on the rationals and completing the rationals with respect to the norm. This gives us a field Qp and we describe its additive structure and the decimal expansion of its members. This leads us into an analysis of p-adic sequences, series, and power series, which concludes with an investigation into the zeros of p-adic power series.

On the passage from local to global in number theory

Bulletin of the American Mathematical Society, 1993

1 Two rational numbers are very close in the p-adic metric if their difference, expressed as a fraction in lowest terms, has the property that its numerator is divisible by a high power of p. The standardly normalized p-adic metric | \p is characterized by the fact that it is multiplicative, that \p\p = \/p , and that if a is an integer not congruent to 0 modp , then \a\p = 1 .

On the p-adic Algebra and its Applications

2009

In this paper we introduce and study the notion of p-adic ring by a complete analogy of that of real ring (see [6]). Then, using this new notion we generalize the concept of p-adic ideal and that of p-adic radical of an ideal (see [17]) to any commutative ring with unit. Afterwards, we illustrate those notions by giving a new characterization of the p-adic spectrum of a ring. Finally, we state and prove an abstract Nullstellensatz of this spectrum.

Metric properties of some special p-adic series expansions

1996

This set Sp is not multiplicatively or additively closed. The function〈 A〉 and set Sp have been used in the study of certain types of p-adic continued fractions by Mahler [14], Ruban [17] and Laohakosol [13] in particular. Recently the fractional part〈 A〉 was used by the present authors [7],[8] to derive some new unique series expansions for any element A∈ Qp, including in particular analogues of certain “Sylvester”,“Engel” and “Lüroth” expansions of arbitrary real numbers into series with rational terms (cf.[16], Chap. IV).

On ppp-rationality of number fields. Applications – PARI/GP programs

Publications Mathématiques de Besançon

On p-rationality of number fields. Applications-PARI/GP programs 2019/2, p. 29-51. http://pmb.cedram.org/item?id=PMB\_2019\_\_\_2\_29\_0 © Presses universitaires de Franche-Comté, 2019, tous droits réservés. L'accès aux articles de la revue « Publications mathématiques de Besançon » (http://pmb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://pmb.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On the metric theory of numbers in non-Archimedean settings

2014

This thesis is a contribution to some fields of the metrical theory of numbers in nonArchimedean settings. This is a branch of number theory that studies and characterizes sets of numbers, which occur in a locally compact topological field endowed with a nonArchimedean absolute value. This is done from a probabilistic or measure-theoretic point of view. In particular, we develop new formulations of ergodicity and unique ergodicity based on certain subsequences of the natural numbers, called Hartman uniformly distributed sequences. We use subsequence ergodic theory to establish a generalised metrical theory of continued fractions in both the settings of the p-adic numbers and the formal Laurent series over a finite field. We introduce the a-adic van der Corput sequence which significantly generalises the classical van der Corput sequence. We show that it provides a wealth of examples of low-discrepancy sequences which are very useful in the quasi-Monte Carlo method. We use our subseq...