Computations in P-Adic Discrete Dynamics and Real Quadratic Fields

Abstract

The field of rational numbers Q is neither complete nor algebraically closed. There is no finite extension of Q which is algebraically closed. Completions of Q with respect to p-adic absolute values are the fields of p-adic numbers Qp. The thesis consists of two parts: p-adic dynamics and real quadratic fields. The first part deals with the p-adic discrete dynamical systems. Two concepts of classical discrete dynamical systems are studied in the context of p-adic fields. Firstly, the notion of topological conjugacy for the p-adic analog of the logistic map and the quadratic map is studied. Secondly, the notion of p-adic backward dynamics for the same maps have also been studied. In the second part the notion of Mersenne primes has been extended to real quadratic fields with class number 1. Computational results are given. The field Q(p2) is studied in detail with a focus on representing Mersenne primes in the form x2 + 7y2. It is also proved that x is divisible by 8 and y ≡ ±3 (mod 8) generalizing the result of F. Lemmermeyer, first proved by H. W. Lenstra and P. Stevenhagen using Artin’s reciprocity law.