There are many attractive and instructive topics which can, and should, be included in an introductory, but moderately ambitious, text book on algebraic number theory. But – as if by a conspiracy of silence – they are usually either omitted altogether, or, at best, are treated inadequately in the existing array of texts available. One of our aims in writing this book has been to try to break free from this standard mould, and to fill these gaps. As instances we mention cubic and biquadratic fields, Gaussian periods, Brauer relations, module theory over a Dedekind domain, an algebraic number theoretic treatment of binary quadratic forms, tame ramification and the two-classgroup of a quadratic field.
Conceptually the book breaks fairly neatly into two parts: the first four chapters and the final chapter are, for the most part, of a theoretical nature, though we always take care to fix abstract ideas by means of worked examples; the remaining three chapters are devoted to giving a detailed study of various arithmetic objects in situations of particular interest.
Throughout the text we have laid great stress on worked examples; it is a depressing fact that many number theorists have never acquired sufficient technique to perform number theoretic calculations in anything but a quadratic field. Again, this is, to some extent, the fault of the existing literature, where scant emphasis is placed on calculations.
On the whole we have opted for schematic exposition, rather than attempting an evolutionary or historical development of the subject matter.