Kurt Foster on Sat, 03 Nov 2012 18:21:49 +0100

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Trace problem

Let K = bnfinit(T)) (T monic and irreducuble in Z[x]) be a number field, R = K.zk its ring of integers, x an 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 of y in R for which trace(Mod(x*y, T)) is zero. But how to determine which units (if any) are in the Z-module spanned by that basis has me flummoxed. Perhaps I am overlooking something obvious.

I suppose you could reduce modulo a rational integer m and look at the units mod mR. That might lead to a proof of nonexistence if no such units exist. Perhaps localization would help?