The $p$-adic Number System and the Artin-Hasse Exponential
Achyut Bharadwaj, Lex Harie Pisco, Krittika Garg, Swayam Chaulagain, Counsellor: Sanskar Agrawal, Mentor: Nischay ReddyJune 2023
$p$-adic numbers,
Artin-Hasse exponential,
radius of convergence,
power series,
integral power series,
discs,
leftist numbers
Introduction #
In this paper, we explore the $p$-adic system, by defining it in multiple ways: as an extension of the $p$-adic integers, as well as an extension of the rationals. We then proceed to perform analysis in the $p$-adics, by defining convergence, continuity and discs. We then describe exponentiation and logarithmic functions over the $p$-adics as functions derived from power series. We explore the radius of convergence and other properties of these functions. We then explore the Artin-Hasse exponential, which, though seemingly random, turns out to be an integral power series.
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