A ?rst introduction to p-adic numbers – Madore, p-adic number – Wikipedia, A ?rst introduction to p-adic numbers – Madore, A ?rst introduction to p-adic numbers – Madore, The p-adic number’s expression through the use of this in nite series is called a p-adic expansion . Each a k for k2fnn+1n+2:::gis known as a p-adic digit. Despite the fact that p-adic expansions are by de nition in nite, they possess an important propert,y making the de nition of Z p concrete and precise. It is discussed in the next proposition.
Relating the base-p periodic expansion of a rational to its p-adic representation, 10/22/2020 · The p-adic Expansion of a Number To construct the p -adic numbers, we define a new norm and a new measure of distance. This new norm appears at first to be strange and artificial, so we will try to motivate it with an example. First, we choose a prime number.
All rationals are periodic in R in decimal expansion . This can be proven by expanding in a geometric sequence. For that same reason, any rational numbers in their expansion in a base are periodic, including the p-adic expansions. We can therefore construct irrational numbers by producing sequences that are aperiodic in its decimal form.
8/14/2019 · 1 Answer1. Active Oldest Votes. 3. For p odd. Z p × = ? p ? 1 × ( 1 + p) Z p where ? p ? 1 = lim n ? ? g p n for g ? Z of order p ? 1 in Z / p Z. If K / Q p is a finite extension whose residue field is O K / ( ?) ? F p f then take g ? O K of order p f ? 1 modulo ( ?) you’ll have ? p f ? 1 = lim n ? ? g p f n.
a p-adic is. 1 First de?nition We will call p-adic digita natural number between 0 and p 1 (inclusive). A p-adic integeris by de?nition a sequence (a i) i2Nof p-adic digits. We write this conventionally as a i a 2 a 1 a 0 (that is, the a iare written from left to right). If nis a natural number, and n= a k 1 a k 2 a 1 a 0, 5/5/2021 · 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).
P-adic Analysis, Field, Rational Number, Number Theory, Hyperreal Number
