Type of Document 
Dissertation 
Author 
Cline, Danny O.

URN 
etd11212004230454 
Title 
On the Computation of Invariants in nonNormal, nonPure Cubic Fields and in Their Normal Closures 
Degree 
PhD 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Parry, Charles J. 
Committee Chair 
Ball, Joseph A. 
Committee Member 
Brown, Ezra A. 
Committee Member 
Haskell, Peter E. 
Committee Member 
Linnell, Peter A. 
Committee Member 

Keywords 
 Cubic Field
 Ideal Class Group
 Normal Closure

Date of Defense 
20041117 
Availability 
unrestricted 
Abstract
Let K=Q(theta) be the algebraic number field formed by adjoining theta to the rationals where theta is a real root of an irreducible monic cubic polynomial f(x) in Z[x]. If theta is not the cube root of a rational integer, we call the field K a nonpure cubic field, and if K doesn't contain the conjugates of theta, we call K a nonnormal cubic field. A method described by Martinet and Payan allows us to construct such fields from elements of a quadratic field. In this work, we examine such nonnormal, nonpure cubic fields and their normal closures, using algorithms in Mathematica to compute various invariants of these fields. In addition, we prove general results relating the ranks of the ideal class groups of the rings of integers of these cubic fields to those of their normal closures.

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