Lectures on finite fields and Galois rings. (English) Zbl 1028.11072

River Edge, NJ: World Scientific. x, 342 p. (2003).
The theory of finite fields and Galois rings has become increasingly important in the last decades. On the one hand there are classical algebraic and number theoretic problems related to this topic and on the other hand it has many applications in computer science, coding theory, and cryptography. In this nice introduction the explicit construction of finite fields and the computation in finite fields are emphasized. (Applications are not included. For a monograph on recent achievements in the theory and applications of finite fields, see I. E. Shparlinski [Finite fields: theory and computation. Dordrecht: Kluwer Academic Publishers (1999; Zbl 0967.11052)]). In particular, the construction of irreducible polynomials and factoring of polynomials over finite fields is discussed. In subsequent chapters quadratic forms over finite fields are analysed and a very good introduction to the theory of Galois rings is given.


11Txx Finite fields and commutative rings (number-theoretic aspects)
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11T06 Polynomials over finite fields
11Y16 Number-theoretic algorithms; complexity
11E04 Quadratic forms over general fields
12E20 Finite fields (field-theoretic aspects)


Zbl 0967.11052