Description

A course in some of the more elementary aspects of algebraic number theory, from a classical perspective. The main strands are continued fractions, binary quadratic forms and modular arithmetic. The theory of continued fractions serves as a unifying theme as well as a source of algorithms. Applications include the representation of primes as sums of squares and the solution of Pell's equation.

Parameters

Syllabus

1. Continued fractions: finite and infinite continued fractions, approximation by rationals, order of approximation.
2. Continued fractions of quadratic surds: applications to the solution of Pell's equation and the sum of two squares.
3. Binary quadratic forms: equivalence, unimodular transformations, reduced form, class number. Use of continued fractions in the indefinite case.
4. Modular arithmetic: primitive roots, quadratic residues, Legendre symbol, quadratic reciprocity. Applications to quadratic forms.

Literature

H Davenport, The Higher Arithmetic, Cambridge University Press (1999).

Allenby & Redfern, Introduction to Number Theory with Computing, Edward Arnold (1989).

Further information

Author: Dr J. Radcliffe Last updated 3 November 1999