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