This monograph provides a self-contained presentation of the foundations
of finite fields, including a detailed treatment of their algebraic
closures. It also covers important advanced topics which are not yet
found in textbooks: the primitive normal basis theorem, the existence of
primitive elements in affine hyperplanes, and the Niederreiter method
for factoring polynomials over finite fields.
We give streamlined and/or clearer proofs for many fundamental results
and treat some classical material in an innovative manner. In
particular, we emphasize the interplay between arithmetical and
structural results, and we introduce Berlekamp algebras in a novel way
which provides a deeper understanding of Berlekamp's celebrated
factorization algorithm.
The book provides a thorough grounding in finite field theory for
graduate students and researchers in mathematics. In view of its
emphasis on applicable and computational aspects, it is also useful for
readers working in information and communication engineering, for
instance, in signal processing, coding theory, cryptography or computer
science.