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:
x12−4x11+76x10+156x9−429x8−2344x7+856x6−2344x5−429x4+156x3+76x2−4x+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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