Kurt Foster on Sat, 03 Nov 2012 18:21:49 +0100
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- To: Pari Users <firstname.lastname@example.org>
- Subject: Trace problem
- From: Kurt Foster <email@example.com>
- Date: Fri, 2 Nov 2012 08:47:31 -0600
- Delivery-date: Sat, 03 Nov 2012 18:21:49 +0100
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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?