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. Language: English