[Paper Review] A brief introduction to p-adic numbers

Stephen Semmes|ArXiv.org|Jan 5, 2003

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TL;DR

This paper provides a concise introduction to p-adic numbers from a classical analyst's perspective, focusing on their arithmetic, ultrametric norm, and completion to form the field ℚₚ. It establishes key properties such as the ultrametric triangle inequality, completeness of ℚₚ, and analogies to complex analysis, including convergence of power series and factorization theorems for entire functions over the algebraic closure ℂₚ.

ABSTRACT

In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex numbers.

Motivation & Objective

• To introduce p-adic numbers and their arithmetic structure to readers with a background in classical analysis.
• To explain how the p-adic absolute value leads to a non-Archimedean metric and completion to ℚₚ.
• To explore the analogy between p-adic analysis and complex analysis, particularly in power series convergence and factorization.
• To demonstrate that ℂₚ, the metric completion of the algebraic closure of ℚₚ, is algebraically closed and supports a factorization theory for entire functions.

Proposed method

• Define the p-adic absolute value |x|ₚ on ℚ by |x|ₚ = p⁻ᵏ for x = pᵏ·m/n with m,n not divisible by p.
• Establish the ultrametric triangle inequality: |x + y|ₚ ≤ max(|x|ₚ, |y|ₚ), which implies |x + y|ₚ ≤ |x|ₚ + |y|ₚ.
• Complete ℚ with respect to |·|ₚ to obtain the field ℚₚ, which is complete and locally compact.
• Extend the p-adic absolute value to the algebraic closure of ℚₚ and complete it to obtain ℂₚ, which is algebraically closed and metrically complete.
• Analyze convergence of power series ∑aₙxⁿ in ℚₚ and ℂₚ, showing convergence iff |aₙ|ₚ|x|ₚⁿ → 0 as n → ∞.
• Establish a factorization theorem for entire functions over ℂₚ: f(x) = c·xᵐ·∏(1 − λⱼx), with zeros at 0 and reciprocals of λⱼ.

Experimental results

Research questions

• RQ1 How does the p-adic absolute value differ from the standard absolute value, and what are its key algebraic and metric properties?
• RQ2 What is the role of the ultrametric triangle inequality in p-adic analysis and the structure of ℚₚ?
• RQ3 How does the completion of ℚ with respect to |·|ₚ yield a complete metric space ℚₚ, and what are its topological features?
• RQ4 In what ways do power series behave differently in ℚₚ compared to ℝ or ℂ, particularly regarding convergence?
• RQ5 What is the structure of the field ℂₚ, and how does it support a factorization theory analogous to that of entire functions in complex analysis?

Key findings

• The p-adic absolute value satisfies the ultrametric inequality |x + y|ₚ ≤ max(|x|ₚ, |y|ₚ), distinguishing it from the Archimedean absolute value.
• The completion of ℚ under |·|ₚ yields the field ℚₚ, which is complete and locally compact, with ℤ ⊂ ℚₚ bounded under |·|ₚ.
• In ℚₚ, a series ∑aₙ converges if and only if |aₙ|ₚ → 0, due to the ultrametric property.
• The algebraic closure of ℚₚ is not complete under |·|ₚ, but its metric completion yields ℂₚ, which is both algebraically closed and metrically complete.
• Every power series ∑aₙxⁿ converging on all of ℚₚ (or ℂₚ) defines a function f(x) that factors as f(x) = c·xᵐ·∏(1 − λⱼx), with finitely many zeros in any bounded region.
• This factorization theorem for entire functions over ℂₚ is analogous to classical Weierstrass factorization in complex analysis, but with simpler convergence conditions due to the ultrametric setting.