Finding minimal polynomials of algebraic numbers of the form tan2(π/n) using Tschirnhausen’s transform

Abstract

Solutions of two problems are proposed based on the Tschirnhausen transform. The first problem is connected with the construction of minimal polynomials of the numbers of the form tan²(π/n) by means of the Tschirnhausen transform for all natural n > 2. The second problem consists in finding the exact roots of the equation x³ − 7x − 7 = 0. A solution of the problem is obtained from the fact that the roots of the equation produce the cyclotomic field Q₇. Examples of construction of minimal polynomials are provided.