Finite Fields: Normal Bases and Completely Free Elements

The central topic of this work is the normal basis theorem, a classical result from field theory. In the last two decades, normal bases in finite fields have been proved to be very useful for doing arithmetic computations. At present, the algorithmic and explicit construction of such bases has become one of the major research topics in finite field theory. Moreover, the search for such bases also led to a better theoretical understanding of the structure of finite fields. In addition to interest in arbitrary normal bases, this volume examines a special class of normal bases whose existence has only been settled more recently. The main problems considered in the present work are the characterization, the enumeration, and the explicit construction of completely free elements in arbitrary finite dimensional extensions over finite fields. Up to now, there is little work on whether the universal property of a completely free element can be used to accelerate arithmetic computations in finite fields. Therefore, the present work belongs to constructive algebra and constitutes a contribution to the theory of finite fields.