Introductory Algebraic Number Theory

Abstract

Algebraic number theory is a subject which came into being through the attempts of mathematicians to try to prove Fermat's last theorem and which now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing and public-key cryptosystems. This book provides an introduction to the subject suitable for senior undergraduates and beginning graduate students in mathematics. The material is presented in a straightforward, clear and elementary fashion, and the approach is hands on, with an explicit computational flavour. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested reading and to the biographies of mathematicians who have contributed to the development of algebraic number theory are given at the end of each chapter. There are over 320 exercises, an extensive index, and helpful location guides to theorems and lemmas in the text.

Euclidean Quotient Rings of Z[√−5]

“…The proof is elementary and there is no reference to any generalized Riemann hypotheses. Throughout this note, the terminology and notations are standard as in [1] or [3]. …”

“…It is well-known that an Euclidean domain is a principal ideal domain (PID), but the converse is not true (see for instance [1], [2]). …”

On Common Roots of Random Bernoulli Polynomials

“…[AW04,Page 93]. These two observations allow us to classify all potential solutions of low algebraic order.…”

Lattices that admit logarithmic worst-case to average-case connection factors

“…Due to lack of space, we will present most facts without proof (which may be found in any number of introductory books on algebraic number theory, e.g. [7,40].) An algebraic number is any root of some polynomial p(x) ∈ Q[x].…”