前者是Archimede绝对值， \\mathbb Q 对其的完备化就是实数域 \\mathbb R ；后者是非Archimede绝对值，满足强三角不等式， \\mathbb Q 对其的完备化就是p-adic数域 \\mathbb Q_p 。这两类域在拓扑上都是完备且局部紧致的。

The p-adic arithmetic allows error-free representation of fractions and error-free arithmetic using fractions. In this tutorial, we describe infinite-precision p-adic arithmetic which is more suitable for software implementations and finite-precision p-adic arithmetic which is more suit-able for hardware implementations.

2025年1月9日 · For any smooth projective curve $Y/K$ of genus $\geq 2$, we prove that the image of the “Selmer” part of Grothendieck’s section set inside the $K_v$-rational points $Y (K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck’s section conjecture.

Any norm j j on Q is equivalent to either the usual norm or a p-adic norm j jp. Recall that we are assuming all of our norms are non-trivial; other-wise the trivial norm would give a third possibility in the statement of the theorem above. Proof. Let a; b > 1 be integers, and write bn in base a as follows: bn = cmam + cm 1am.

The p-adic valuation allows defining an absolute value on p-adic numbers: the p-adic absolute value of a nonzero p-adic number x is | x | p = p − v p ( x ) , {\displaystyle |x|_{p}=p^{-v_{p}(x)},} where v p ( x ) {\displaystyle v_{p}(x)} is the p -adic valuation of x .

2024年1月15日 · I show that rising-continuous (p,q)-adic functions are measurable, which we then use to show that they induce q-adic-valued random variables (RVs) on Z_p. I then prove a case of Parseval identity in the form of an equality between the ell 2 norm of our RV’s characteristic function (phi_chi) and the sum of the squares of the RV’s probability ...

n) be a p-adic integer with each a nuniquely represented by an integer in 2[0;pn 1]. The sequence (b 0;b 1;b 2;:::) with b 0 = a 1 and b n= (a n+1 a n)=pn is called the p-adic expansion of a. Theorem 4.6. Every element of Z p has a unique p-adic expansion and every sequence (b 0;b 1;b 2;:::) of integers in [0;p 1] is the p-adic expansion of an ...

We start this chapter by introducing p-adic integers, both intuitively by referring to writing an integer in a given base p, and formally by defining the concept of inverse limit. This latter approach will allow to show that p-adic integers form a ring, denoted by Zp. We will then consider ”fractions” of p-adic integers, that

We now indicate one way to get a handle on p-adic numbers for the purpose of computation, inspired by the theory of decimal expansions for elements of R. Proposition 0.10.

Proposition. The p-adic norm is non-archimedean, that is, |a+b| p≤max{|a| p,|b| p}. Proof. First, write the numbers in reduced form, a= pv p(a)a′and b= pv p(b)b′ where aand bare rational numbers with both numerator and denominators not containing multiples of p. Now, because v p(a),v p(b) ≥min{v p(a),v p(b)}we can write a+b= pmin{v p(a),v