On Browkin's conjecture about the elements of order five in \(K_2(\mathbb Q)\) (Q2372581)

Based on the work of Lenstra (letter to Browkin, May 1981), a succinct proof of Browkin's conjecture that the elements of order five in \(K_2(\mathbb Q)\) can be written as a product of elements of the form \(\{a, \Phi_5(a)\}\), where \(a\in\mathbb Q^*\), is given. Here, as usual, \(K_2\) denotes Milnor's group and \(\Phi_n\) is the \(n\)th cyclotomic polynomial. However, the author only makes use of a classical result of \textit{C. S. Davis} [Acta Math. 84, 263--298 (1951; Zbl 0042.04502)] on binary quartic forms instead of Zantema's idea and van der Geer's results used in Lenstra's proof. It is stated that a similar method can be used to prove similar results on the elements of order ten or twelve in \(K_2(\mathbb Q)\).