Finite Fields - Hachenberger, Dirk

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Résumé :
The central topic of emFinite Fields: Normal Bases and Completely Free/em emElements/em is the famous 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. br/ In addition to interest in arbitrary normal bases, emFinite Fields:/em emNormal Bases and Completely Free Elements/em 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 no work done stating 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. br/ This book serves as an excellent reference for researchers in finite fields, and may be used as a text for advanced courses on the subject.

Sommaire:
Dirk Hachenberger is a mathematician working in the fields of combinatorics, number theory, applicable algebra, finite geometry and coding theory. He is the recipient of the 2004 Hall medal of the Institute for Combinatorics and its Applications (ICA). He was also awarded the price for good teaching of the Bavarian State Minister for Science, Research and Arts in 2004. He has published the books Finite Fields: Normal bases and completely free elements (in English) and Mathematics for computer scientists (in German). Dieter Jungnickel is an internationally known mathematician working in the fields of applicable algebra, coding theory, design theory, finite geometry, combinatorics and combinatorial optimization. He is the recipient of the 2018 Euler medal of the Institute for Combinatorics and its Applications (ICA). He has published several well-known books, including Design Theory, Optimization Methods, Finite fields: Structure and arithmetics, CodingTheory, Combinatorics and Graphs, Networks and Algorithms, some of which have been published both in English and German....