Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K_d/\mathbb Q$ of degree d ramified only at p. It is contained in the cyclotomic field $\mathbb Q(\zeta)$, where $\zeta$ is a primitive p-th root of unity. $K_d$ is generated over $\mathbb Q$ by the Gaussian period $\omega = \operatorname{Tr}_{~\mathbb Q(\zeta)/K} (\zeta)$ (see Gupta,Zagier).
Now there are results concerning the higher coefficients of the minimal polynomial of $\omega$ over the rationals, my question is: what can be done for the constant term of the minimal polynomial, could someone point out results in this direction.
