A bound on the index of exponent-4 algebras in terms of the u-invariant

For a prime number p, an integer e >= 2 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.