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

For a prime number $p$, an integer $e\ge 2$ 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.