On p-adic Continued Fractions and Quadratic Irrationals

Abstract

In this dissertation we investigate prior definitions for p-adic continued fractions and introduce some new definitions. We introduce a continued fraction algorithm for quadratic irrationals, prove periodicity for Q₂ and Q₃, and numerically observe periodicity for Q(p) when p < 37. Various observations and calculations regarding this algorithm are discussed, including a new type of symmetry observed in many of these periods, which is different from the palindromic symmetry observed for real continued fractions and some previously defined p-adic continued fractions. Other results are proved for p-adic continued fractions of various forms. Sufficient criteria are given for a class of p-adic continued fractions of rational numbers to be finite. An algorithm is given which results in a periodic continued fraction of period length one for √D ∈ Zˣ(p), D ∈ Z, D non-square; although, different D require different parameters to be used in the algorithm. And, a connection is made between continued fractions and de Weger’s approximation lattices, so that periodic continued fractions can be generated from a periodic sequence of approximation lattices, for square roots in Zˣ(p). For simple p-adic continued fractions with rational coefficients, we discuss observations and calculations related to Browkin’s continued fraction algorithms. In the last chapter, we apply some of the definitions and techniques developed in the earlier chapters for Q(p) and Z to the t-adic function field case F(q)((t)) and F(q)[t], respectively. We introduce a continued fraction algorithm for quadratic irrationals in F(q)((t)) that always produces periodic continued fractions.