Saturday, 31 July 2010

Unit in a number field with same absolute value at a real and a complex place

I was asked whether it was possible to produce a monic polynomial with integer coefficients, constant coefficient equal to 1, having a real root r>1 and a pair of complex roots with absolute value r, which are not r times a root of unity. Bonus if the polynomial did not have roots of absolute value one. An answer (without the bonus) is:



x124x11+76x10+156x9429x82344x7+856x62344x5429x4+156x3+76x24x+1.



I'd like an answer to the bonus question in the following strengthened form: Is there a unit r in a number field such that r has the same absolute value (bigger than one) at a real and a complex place (of mathbbQ(r) to avoid trivial answers) but no archimedian place where
r has absolute value 1?

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