Sinposis

"New to the Fifth EditionPell's Equation x^2-dy^2=1: all solutions can be obtained from a single `fundamental' solution, which can be found using continued fractions.Galois theory of number field extensions, relating the field structure to that of the group of automorphisms.More material on cyclotomic fields, and some results on cubic fields.Advanced properties of prime ideals, including the valuation of a fractional ideal relative to a prime ideal, localisation at a prime ideal, and discrete valuation ringsRamification theory, which discusses how a prime ideal factorises when the number field is extended to a larger one.A short proof of the Quadratic Reciprocity Law based on properties of cyclotomic fields. This Valuations and p-adic numbers. Topology of the p-adic integers."