Date of Award

1986

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Abstract

This dissertation is concerned with dyadic ramification in quartic number fields E. If f(x) is a defining polynomial for E, then E (TURNEQ) IQ x /(f(x)). The problem is to decide whether or not 2 ramifies in E, in terms of the coefficients of f(x). In this dissertation, this problem is. studied when f(x) is an irreducible quartic trinomial. That is, f has one of the forms (UNFORMATTED TABLE FOLLOWS). (1) f(x) = x('4) + cx + d. (2) f(x) = x('4) + cx('2) + d c,d in ZZ. (TABLE ENDS). (any irreducible trinomial can be reduced to one of these two forms). When f(x) has the first form f(x) = x('4) + cx + d. I have shown that 2 is always ramified in E except possibly when the polynomial discriminant d(,f) = 5 (.) t('2) iin the 2-adic integers ZZ(,2). When f(x) has the form f(x) = x('4) + cx('2) + d. I use Weil's additive characters (gamma)(,p) of the rational Witt ring W(IQ) to devise a test of whether or not 2 ramifies in E. The result appears as Theorem 2. Applications of Theorem 2 to fundamental units in real quadratic fields are discussed.

Pages

63

Share

COinS