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16
TwoDimensional Topological Quantum Field Theories And Frobenius Algebras
 J. Knot Theory Ramifications
, 1996
"... We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either ..."
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We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either "annihilator algebras"  algebras whose socle is a principal ideal  or field extensions. The relationship between twodimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable twodimensional topological quantum field theories. Keywords: topological quantum field theory, frobenius algebra, twodimensional cobordism, category theory 1. Introduction Topological Quantum Field Theories (TQFT's) were first described axiomatically by Atiyah in [1]. Since then, much work has been done ...
A database of local fields
 J. Symbolic Comput
"... 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. 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.
Normal Bases over Finite Fields
, 1993
"... 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 repr ..."
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Cited by 12 (0 self)
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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...
CLASSES OF FORMS WITT EQUIVALENT TO A SECOND TRACE FORM OVER FIELDS OF CHARACTERISTIC TWO
, 2004
"... 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. ..."
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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.
ON PAC EXTENSIONS AND SCALED TRACE FORMS
, 2007
"... In this work we address the question: when is a quadratic form isomorphic to a scaled trace form. We give an exact answer over pseudo algebraically closed fields. We also show that for solvable extensions of a countable Hilbertian field any nondegenerate quadratic form of degree at least 5 is isomo ..."
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In this work we address the question: when is a quadratic form isomorphic to a scaled trace form. We give an exact answer over pseudo algebraically closed fields. We also show that for solvable extensions of a countable Hilbertian field any nondegenerate quadratic form of degree at least 5 is isomorphic to a scaled trace form.
ON PAC EXTENSIONS AND SCALED TRACE FORMS
, 2007
"... Any nondegenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and primetop extensions of a Hilbertian field. The proofs are based on the theory of PAC ext ..."
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Any nondegenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and primetop extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.
TRACE FORMS OF GALOIS EXTENSIONS IN THE PRESENCE OF A FOURTH ROOT OF UNITY
, 2003
"... Abstract. We study quadratic forms that can occur as trace forms qL/K of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that qL/K is always a scaled Pfister form. We prove this conjecture and classify the finite groups G whic ..."
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Abstract. We study quadratic forms that can occur as trace forms qL/K of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that qL/K is always a scaled Pfister form. We prove this conjecture and classify the finite groups G which admit a GGalois extension L/K with a nonhyperbolic trace form. We also give several applications of these results. 1.