Finite Fields: Normal Bases and Completely Free Elements / Edition 1

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The central topic of Finite Fields: Normal Bases and Completely Free Elements 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. In addition to interest in arbitrary normal bases, Finite Fields: Normal Bases and Completely Free Elements 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. 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.

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Editorial Reviews

A research reference and text examining the Normal Basis Theorem, the classical result from field theory. Hachenberger U. of Augsburg, Germany details a special class of normal bases whose existence has only been recently settled, considering the main problems of characterization, the enumeration, and the explicit construction of complete free elements in arbitrary finite dimensional extensions over finite fields. Although the universal property of a completely free element used to accelerate arithmetic computation in finite fields has not been ascertained, this volume represents the search for such elements and leads to a deeper insight of the finite fields structure. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Table of Contents

Preface. I: Introduction and Outline. 1. The Normal Basis Theorem. 2. A Strengthening of the Normal Basis Theorem. 3. Preliminaries on Finite Fields. 4. A Reduction Theorem. 5. Particular Extensions of Prime Power Degree. 6. An Outline. II: Module Structures in Finite Fields. 7. On Modules over Principal Ideal Domains. 8. Cyclic Galois Extensions. 9. Algorithms for Determining Free Elements. 10. Cyclotomic Polynomials. III: Simultaneous Module Structures. 11. Subgroups Respecting Various Module Structures. 12. Decompositions Respecting Various Module Structures. 13. Extensions of Prime Power Degree (1). IV: The Existence of Completely Free Elements. 14. The Two-Field Problem. 15. Admissibility. 16. Extendability. 17. Extensions of Prime Power Degree (2). V: A Decomposition Theory. 18. Suitable Polynomials. 19. Decompositions of Completely Free Elements. 20. Regular Extensions. 21. Enumeration. VI: Explicit Constructions. 22. Strongly Regular Extensions. 23. Exceptional Cases. 24. Constructions in Regular Extensions. 25. Product Constructions. 26. Iterative Constructions. 27. Polynomial Constructions. References. List of Symbols. Index.

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