Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...

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Abstract. We describe our online database of finite extensions of Q p, and how it can be used to facilitate local analysis of number fields. 1.

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Abstract. Let F be a field of characteristic two. We determine all nonhyperbolic quadratic forms over F that are Witt equivalent to a second trace form. 1.

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This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is an exposition of a family of “generic ” polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of “generic dimension” to address the problem of the smallest number of parameters required by a generic polynomial. Mathematical Sciences Research Institute

Abstract. Let 2 + S4 denote the double cover of S4 corresponding to the element in H2 (S4, Z/2Z) where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4. Given an elliptic curve E, letE[2] denote its 2-torsion points. Under some conditions on E elements in H1 (GalQ,E[2])\{0} correspond to Galois extensions N of Q with Galois group (isomorphic to) S4. Inthisworkwegiveaninterpretation of the addition law on such fields, and prove that the obstruction for N having a Galois extension Ñ with Gal(Ñ/Q) ≃ 2 + S4 gives a homomorphism s + 4: H1 (GalQ,E[2]) → H2 (GalQ, Z/2Z). As a corollary we can prove (if E has conductor divisible by few primes and high rank) the existence of 2-dimensional representations of the absolute Galois group of Q attached to E and use them in some examples to construct 3/2 modular forms mapping via the Shimura map to (the modular form of weight 2 attached to) E.

First and foremost, I would like to thank the Trustees of the University of Pennsylvania and the entire Mathematics Department for putting so much faith in my success. Without providing funding for my five years here, including first- and fourth-year fellowships and a Calabi Fellowship, this project would certainly not have been possible. I also sincerely thank the School of Arts and Sciences for granting me a Dissertation Completion Fellowship for the Fall semester of 2008. I am perhaps even more indebted to my advisor, Ted Chinburg, at the University of Pennsylvania. Giving me enough support and space to chart my own way through the tangle of questions that eventually coalesced into my thesis, Ted has been a great mentor, an astute advisor, and a cherished friend. Aside from doing math together, Ted and I have: looked through telescopes, shopped for meat cleavers in Chinatown, done laundry, wandered around aimlessly, cooked steaks, convinced ourselves that we were in grave danger of contacting Lyme disease, and laughed a lot. Ted is a master of making things sound interesting, no matter the odds, as well as seeing the big picture so clearly that almost anything can fit in. I would also like to thank the professors at Penn who have had such an important impact on my mathematical developement. I first learned Galois cohomology and about the Brauer group

Abstract. We show that the symmetric track group Sym □ (n), which is an extension of the symmetric group Sym(n) associated to the second Stiefel-Withney class, acts as a crossed module on the secondary homotopy group of a pointed space.