Introduction

Throughout, the reader is assumed to already have a good (working) knowledge of basic undergraduate algebra. For ease of reference, we shall first recall some central definitions and useful elementary results concerning groups, rings, fields, modules, vector spaces, and algebras. For additional basic theory, full details and more, please refer, for example, to Serge Lang's Algebra [126].

Furthermore, at some points we shall need some results from field theory and algebraic number theory, as well as from multilinear algebra, specifically concerning cyclotomic number fields and tensor products. In each of these cases we shall give a brief introduction and state the required results. For full details and more, please refer to references 125, 126, 127. Our exposition of these topics (loosely) follows Lang [126], except where stated otherwise.

Finally, we shall need results from the theory of algebraic function fields (in one variable) over finite fields, that is, algebraic curves over finite fields. This is deferred to Section 12.7. Specifically, the focus there will be on families of curves with asymptotically many rational points. We will give a bird's-eye introduction to this topic that is self-contained in that it assumes as background only the material on basic algebra covered earlier on. For a full treatment of the basic theory of algebraic function fields, as well as such results as the ones referred to earlier, we refer to Henning Stichtenoth's Algebraic Function Fields and Codes [172]. Except when stated differently, our exposition follows Stichtenoth [172], specifically parts of Chapters 1, 3, 5, and 7. Full proofs are mostly beyond the scope of this introduction. Sometimes a sketch is given. Yet we shall state all results needed for our purposes.

Basic definitions and results that are covered in full detail by most standard introductory textbooks are mentioned without reference (but sometimes a proof is given). In case we suspect that a result we quote is not so easy to look up elsewhere in full detail (say, if it is disguised as a special case of more advanced theory or hidden in exercises), we give an explicit reference to a suitable text.

In a few places in the text we will give appropriate references when discussing certain specific (advanced) algorithmic aspects.