Finite Fields

Finite Fields are fundamental structures of Discrete Mathematics. They serve as basic data structures in pure disciplines like Finite Geometries and Combinatorics, and also have aroused much interest in applied disciplines like Coding Theory and Cryptography. A look at the topics of the proceed ings volume of the Third International Conference on Finite Fields and Their Applications (Glasgow, 1995) (see [18]), or at the list of references in I. E. Shparlinski's book [47] (a recent extensive survey on the Theory of Finite Fields with particular emphasis on computational aspects), shows that the area of Finite Fields goes through a tremendous development. The central topic of the present text is the famous Normal Basis Theo rem, a classical result from field theory, stating that in every finite dimen sional Galois extension E over F there exists an element w whose conjugates under the Galois group of E over F form an F-basis of E (i. e. , a normal basis of E over F; w is called free in E over F). For finite fields, the Nor mal Basis Theorem has first been proved by K. Hensel [19] in 1888. Since normal bases in finite fields in the last two decades have been proved to be very useful for doing arithmetic computations, at present, the algorithmic and explicit construction of (particular) such bases has become one of the major research topics in Finite Field Theory.

Klappentext

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.

Inhalt
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. Admissability.- 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.