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Gauss and p-adic numbers

Franz Lemmermeyer

Abstract

In his notebooks, Gauss recorded various calculations with "infinite congruences". These infinite congruences are p-adic numbers; Gauss computes a square root of $5$ in the $11$-adic integers in order to find an $11$-adic approximation to a quadratic Gauss sum, computes a nontrivial square root of $1$ in $10$-adic integers, and computes the $10$-adic logarithms of small natural numbers.

Gauss and p-adic numbers

Abstract

In his notebooks, Gauss recorded various calculations with "infinite congruences". These infinite congruences are p-adic numbers; Gauss computes a square root of in the -adic integers in order to find an -adic approximation to a quadratic Gauss sum, computes a nontrivial square root of in -adic integers, and computes the -adic logarithms of small natural numbers.

Paper Structure

This paper contains 4 sections, 34 equations, 7 figures.

Table of Contents

  1. $p$-adic roots of polynomials
  2. $p$-adic approximations of quadratic Gauss sums
  3. Square roots of $1$ in $10$-adic numbers
  4. Logarithms in $10$-adics

Figures (7)

  • Figure 1: Infinite Congruence
  • Figure 2: Square root of $5$ in the $11$-adic numbers
  • Figure 3: Computation of an $11$-adic approximation of $\sqrt{5}$
  • Figure 4: Computation of $11$-adic approximations of quadratic periods
  • Figure 5: Calculation of a $10$-adic square root of $1$
  • ...and 2 more figures