LSU Historical Dissertations and Theses

1986

Dissertation

Degree Name

Doctor of Philosophy (PhD)

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.

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