For a prime number p, an integer e > 1 and a field F containing a primitive p^e-th root of unity, the index of central simple F-algebras of exponent p^e is bounded in terms of the p-symbol length of F. For a nonreal field F of characteristic different from 2, the index of central simple algebras of exponent 4 is bounded in terms of the u-invariant of F. Finally, a new construction for nonreal fields of u-invariant 6 is presented.

We thank the referee for their comments. This work was supported by the Fonds Wetenschappelijk Onderzoek-Vlaanderen (FWO) in the FWO Odysseus Programme (project G0E6114N 'Explicit Methods in Quadratic Form Theory'), the Fondazione Cariverona in the programme Ricerca Scientifica di Eccellenza 2018 (project 'Reducing complexity in algebra, logic, combinatorics - REDCOM'), and by the FWO-Tournesol programme (project VS05018N).