To make this text as self-contained as possible, we give a brief but basically self-contained sketch of the theory of algebraic number fields in §1.2. We also summarize necessary facts from linear (and homological) algebra in §1.1 and from the theory of p-adic numbers in §1.3. For a first reading, if the reader has basic knowledge of these subjects, he or she may skip this chapter and consult it from time to time as needed in the principal text of the book. We suppose in §1.2 basic knowledge of elementary number theory, concerning rational numbers and algebraic numbers, which is found in any standard undergraduate level text. We shall concentrate on what will be used in the later chapters. Readers who want to know more about algebraic number theory should consult [Bourl,3], [FT], [Wl] and [N].

Linear algebra over rings

We summarize in this section some facts from linear algebra and some from homological algebra. We will not give detailed proofs.

Let A be a commutative ring with identity. For two A-modules M and N, we write HomA(M,N) for the A-module of all A-linear maps of M into N. In particular, M* = HomA(M,A) is called the A-dual module of M. A sequence of A-linear maps M N L is called ”exact〈 (at N) if Im(α) = Ker(β). A sequence …→ Mi-1 → Mi → Mi+1 →… is called exact if it is exact at Mi for every i.