Kurt Foster on Sat, 10 Nov 2012 16:08:26 +0100

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Trace problem

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.