p-adic number - Wikipedia The 3-adic integers, with selected corresponding characters on their Pontryagin dual group In number theory, given a prime number p, [note 1] the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; p -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime
p-adic Number -- from Wolfram MathWorld A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric " Any nonzero rational number x can be represented by x=(p^ar) s, (1) where p is a prime number, r and s are integers not divisible by p, and a is a unique integer Then define the p-adic norm of x by |x|_p=p^(-a) (2) Also
Introduction to p-adic Numbers - Stanford University were we have plugged in x = 2 Now, your calculus teacher may object that this formula is only defined for |x| < 1 but if we listened to every naysayer screaming that our ideas don’t quite make sense how far can we really get? In this class we are going to consider an alternative to the real numbers in which this formula makes sense First, we need to think about what an infinite sum really
P-ADIC NUMBERS, QUADRATIC FORMS, AND THE HASSE-MINKOWSKI THEOREM Abstract In this paper, we will explore the Hasse-Minkowski theorem and the local-global principle in number theory We will introduce the field of p-adic numbers as well as the notions of quadratic forms, local and global fields, and Hilbert symbols We shall also cover the topics of Hensel’s lemma, the approximation theorems, and the Hilbert reciprocity Eventually, we will prove the
p-adic Numbers: An Introduction | SpringerLink This textbook provides an introduction to p-adic numbers suitable for upper undergraduate students It covers valuations, Hensel's lemma, p-adic analysis, and field extensions The book teaches the use of open source software Sage or GP, with lots of examples and exercises included throughout