ABSTRACT
This chapter deals with finite field algebra particularly useful for various channel coding and cryptography techniques. It describes operations in the field of real numbers R or rational numbers Q with the usual notions of addition, multiplication, subtraction and division. For the case of finite fields the operations and properties are quite different; hence, it needs to go back to the basics and define these algebraic structures from the first principles. The chapter also describes the general definitions of groups, rings, and fields. It discusses finite fields and some of the properties and focuses on shifts to the extension fields of the binary field and polynomials with binary coefficients. The chapter highlights a couple of applications of the theory and the connection to the theory of Galois fields and considers a specific case with 255 bytes as codewords correcting t many byte errors.
