Monograph of famous mathematicians, reflecting the numerous connections of the classical branch of algebra - the theory of finite fields - with combinatorics, coding theory, and the theory of automata. The presentation is distinguished by simplicity and clarity, a large number (about 600) of examples and exercises. For applied mathematicians, research engineers, graduate students and university students. Volume 1: Chapters 1-6.Volume 2: Chapters 7-10.