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Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith
Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith
LSU Doctoral Dissertations
This dissertation is concerned with providing a description of certain symmetric bilinear forms, called trace forms, associated with finite normal extensions N/K of an algebraic number field K, with abelian Galois group Gal(N/K). These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). Complete descriptions are obtained when the base field K has exactly one dyadic prime and either no real embeddings or one real embedding. For these fields K, the set of abelian trace forms is closed under multiplication in the Witt ring W(K).
Exotic Integral Witt Equivalence Of Algebraic Number Fields, Changheon Kang
Exotic Integral Witt Equivalence Of Algebraic Number Fields, Changheon Kang
LSU Doctoral Dissertations
Two algebraic number fields K and L are said to be exotically integrally Witt equivalent if there is a ring isomorphism W(OK) ~ W(OL) between the Witt rings of the number rings OK and OL of K and L, respectively. This dissertation studies exotic integral Witt equivalence for totally complex number fields and gives necessary and sufficient conditions for exotic integral equivalence in two special classes of totally complex number fields.
Bounding The Wild Set (Counting The Minimum Number Of Wild Primes In Hilbert Symbol Equivalent Number Fields), Marius M. Somodi
Bounding The Wild Set (Counting The Minimum Number Of Wild Primes In Hilbert Symbol Equivalent Number Fields), Marius M. Somodi
LSU Doctoral Dissertations
This dissertation makes a contribution to the study of Witt rings of quadratic forms over number fields. To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. The situation is particularly nice when this number is 0; often it is not 0. Earlier investigations have established lower bounds for this number. In this dissertation an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered …