Integral basis, discriminant and indices in cubic fields

INTEGRAL BASIS, DISCRIMINANT AND INDICES IN CUBIC FIELDS (M.Sc. Thesis) Mustafa YAPICI GAZI UNIVERSITY INSTITUTE OF SCIENCE AND TECHNOLOGY September 1988 ABSTRACT In this study the field extension from n. degree of the Q rational number field has been taken into consideration and being made calculations with numbers in cubic field which have a position (n = 3), algebraic integer in the field and ring which they have formed, have been examined. Making use of norm and trace defini tions, the discriminant of it has been defined and some theorems related to its discriminant that we used in the third part have been proved. Being brought Galois group as a reply to some finite extensions, the relationship between Galois group on F field of f, GalF(f) and the discriminant of f polynomial and the relationship between discriminant of field and field have been examined. Being given a definition of index the integral basis of Q((ab2)1/3 ) pure cubic field have been given, existence of a cubic field whose minimum index is greater than any previously assigned N positive number. -Ill-