Indices of algebraic integers in cubic fields

Abstract

Let F = Q(¿) be a cubic field with ¿ ¿ OF . The index of ¿ in OF is the Z-module index ind(¿) := [OF : Z[¿]] ¿ N. The minimal index of F is given by m(F) = min ¿¿OF ind(¿). Let d ¿ Z be squarefree. If d 6= 1, let C(d) denote the set of all non-cyclic cubic fields whose normal closure contains the unique quadratic subfield Q( v d). Let C(1) denote the set of all cyclic cubic fields. For a given squarefree d ¿ Z, we determine the set of all index values assumed by algebraic integers in cubic fields in each subfamily of C(d) with a given factorization of the prime ideal (2). We also determine that each index assumed is assumed by infinitely many algebraic integers in distinct cubics fields within this subfamily. Moreover, for each N ¿ N, we show that there exists a cubic field F ¿ C(d) with m(F) N.