Authors’ abstract: We prove a version of Schanuel’s theorem [Heights in number fields, Bull. Soc. Math. Fr. 107, 433-449 (1979; Zbl 0428.12009)] in the noncommutative case: we provide an asymptotic formula for the number of one-dimensional left subspaces of \(D^N\) of height at most \(H\), where \(D\) is a finite dimensional rational division algebra, \(N\) a positive integer and \(H\) a real number. The height, as considered in a previous paper, is defined with the help of a maximal order in \(D\) and a positive anti-involution. We give a completely explicit main term involving class number, regulator, discriminant and zeta function of \(D\). We also compute an explicit error term.