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: 
$x^{12} - 4x^{11} + 76x^{10} + 156x^9 - 429x^8 - 2344x^7 + 856x^6 - 2344x^5 - 429x^4 + 156x^3 + 76x^2 - 4x + 1$.
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 $\mathbb{Q}(r)$ to avoid trivial answers) but no archimedian place where 
$r$ has absolute value $1$?
 A: I got an answer from Jeff Vaaler, whose office is next door to mine. Sometimes the internet is not the best source. Although, since he didn't remember the author, I wouldn't have located the paper without mathscinet.
It's impossible to get the "strenghtened form". It follows from the results of:
Ferguson, Ronald,
Irreducible polynomials with many roots of equal modulus.
Acta Arith. 78 (1997), no. 3, 221--225. 
A: The polynomial $f(x)=x^3+3x^2+2x-1=(x+1)^3-(x+1)-1$ has a real root $q$, $0 < q < 1$, and complex conjugate roots $s$ and $t$, with $st=1/q=r^2$, with $r=|s|=|t|>1$. The polynomial 
$(x^2-1/q)(x^2-1/s)(x^2-1/t)=x^6-2x^4-3x^2-1$ has $r$ as one of its roots. So I think the product 
$(x^3+3x^2+2x-1)(x^6-2x^4-3x^2-1)$ should work. 
What have I missed?
A: The big problem is that Abs(a) = b implies that a = b x 1c.  Whether or not c is a rational number is another question, however.  If I gave you $x^3 - 11x^2 + 55x -125$, its roots are x = 5, x = 3+4*i*, x = 3-4*i*  In this case, |3+4*i*| = 5, and 5*(-1)ArcTan(3/4)/Pi = 3+4*i*, and ArcTan(3/4)/Pi evaluates to ~ .295...  Granted, this doesn't satisfy the requirement that the constant is equal to 1, but the point still stands that any two numbers which share an absolute value only have a factor difference of some power of unity.
