Polynomials generating maximal real subfields of circular fields

We have constructed recurrence formulas for polynomials $q_n(x)\in\mathbb Q[x]$, any root of which generates the maximal real subfield of circular field $K_{2n}$. It has been shown that all real subfields of fixed field $K_{2n}$ can be described by using polynomial $q_n(x)$ and its Galois group. Furthermore, a methodology has been developed for presentation of square radical $\sqrt d$, $d\in\mathbb N$, $d>1$ in the form of a polynomial with rational coefficients relative to $2\cos(\pi/n)$ at the corresponding $n$. The theoretical results have been verified by a number of examples.