## LSU Historical Dissertations and Theses

1989

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Robert V. Perlis

#### Abstract

The author studies reciprocity equivalence and the wild set of a reciprocity equivalence. He proves that if two algebraic number fields K and L are reciprocity equivalent then there exists a reciprocity equivalence between them with an infinite wild set. In particular, there always exists a self-equivalence with an infinite wild set on any algebraic number field. Even though a wild set of an equivalence can be infinite, he proves that its Dirichlet density is always zero. The structure of a reciprocity equivalence is examined. He proves that the bijection on primes determines the group isomorphism on global square classes in an equivalence. Indeed he proves more. Namely, if (t\$\sb1,T\sb1\$) and (t\$\sb2,T\sb2\$) are reciprocity equivalences from K to L and the bijections T\$\sb1\$ and T\$\sb2\$ on primes agree on a set of Dirichlet density bigger than zero, then the global square class group isomorphisms t\$\sb1\$ and t\$\sb2\$ agree everywhere, and T\$\sb1\$ agrees with T\$\sb2\$ at every noncomplex prime.

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