Polynomials of Galois groups [2^n]S(n) for which the real part of the roots is rational

Hi all

there are irreducible polynomials of degree 2*n
with Galois group [2^n]S(n) for which the real part
of the some or all roots are rational.

Do the imaginary parts satisfy a special relationship as well in this
case?

The following polynomial of degree 12,
has 4 roots where the real part is -1:

p=x^12 + 12*x^11 + 132*x^10 + 880*x^9 + 3960*x^8 + 12672*x^7 +
29568*x^6 + 50688*x^5 + 63360*x^4 + 56320*x^3 + 33792*x^2 + 12288*x +
2814

The roots:
x[1]=-0.2201556847 - 0.4769078063*I
x[2]=-0.2201556847 + 0.4769078063*I
x[3]=-1.000000000 - 7.595754021*I
x[4]=-1.000000000 + 7.595754021*I
x[5]=-1.000000000 - 2.418443252*I
x[6]=-1.000000000 + 2.418443252*I
x[7]=-1.312804678 - 1.306288410*I
x[8]=-1.312804678 + 1.306288410*I
x[9]=-0.6871953223 - 1.306288410*I
x[10]=-0.6871953223 + 1.306288410*I
x[11]=-1.779844315 - 0.4769078063*I
x[12]=-1.779844315 + 0.4769078063*I

And the sum of the roots:

x[1]+x[11]=-2.000000000 - 0.9538156126*I
x[2]+x[12]=-2.000000000 + 0.9538156126*I
x[7]+x[ 9]=-2.000000000 - 2.612576820*I
x[8]+x[10]=-2.000000000 + 2.612576820*I
x[1]+x[11]=-2.000000000 - 0.9538156126*I

any comments are welcome.

Gerry
.

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