Summary: The aim of this paper is to show that the simplest techniques of linear algebra allow us to make explicit the defining equations of the maximal real cyclotomic extensions \(\mathbb Q(\zeta+\zeta^ {-1})\) of \(\mathbb Q(\zeta)\), where \(\zeta\) stands for a primitive \(p^ \nu\)-th root of unity with \(p\) a rational prime and \(\nu\) any positive integer.