Trace problem

[Kurt Foster <drsardonicus@earthlink.net>]

Let K = bnfinit(T)) (T monic and irreducuble in Z[x] of degree n > 1,  
with Galois group cyclic of order n) be a number field, R = K.zk its  
ring of integers, Mod(x,T) a non-zero element of R.  I want to  
determine all units u such that trace(Mod(x*u,T)) is zero.

It's certainly possible to obtain a Z-basis of the set Y of Mod(y,T)  
in R for which

trace(Mod(x*y, T)) = 0.

But how to determine which units (if any) are in Y has me flummoxed.  
Perhaps I am overlooking something obvious.  I note that Y only has  
rank n-1, one less than the rank n of R.  If memory serves, Borevich  
and Shafarevich's NUMBER THEORY covers the subject of "non-full  
modules" to some extent.

These units arise in the my paper "HT90 and `simplest number  
fields" (abstract and links to preprint at http://arxiv.org/abs/1207.6099) 
.  When n = 3, the cyclic cubics having such units as zeroes are  
Shanks's simplest cubics.  When n = 4, the cyclic quartics having such  
units as zeroes are a family which includes the usual defining  
polynomials for the "simplest" quartic fields and e family of  
alternate defining polynomials for Washington's cyclic quartic fields  
as special cases.  The "simplest" number fields of degrees 5 and 6  
also contain such units.