Determination of the minimal polynomials of algebraic numbers of the form $\operatorname{tg}^2(\pi/n)$ by the Tschirnhausen transformation

Solutions of two problems are offered based on the Tschirnhausen transformation. The first problem is connected with the construction of minimal polynomials of the numbers of the form $\operatorname{tg}^2(\pi/n)$ by means of the Tschirnhausen transformation for all natural $n>2$. The second problem consists in finding the exact values of the roots of the equation $x^3-7x-7=0$. The solution of the problem is obtained by considering the fact that the roots of the equation produce the circular field $\mathbb Q_7$. The examples of the construction of minimal polynomials are provided.