Strong u-invariant and period-index bound for complete ultrametric fields

Let k be a complete ultrametric field with ${\text{dim}}_Q\left(\sqrt{\left|k^{⁎}\right|}\right)=n$ finite, with residue field $\tilde{k}$, and char$\left(\tilde{k}\right)\ne 2$. We prove that $u\left(k\right)\le 2^{n}u\left(\tilde{k}\right)$. Let $u_s\left(k\right)$ be the strong u-invariant of k, then we further show that $u_s\left(k\right)\le 2^n{u}_s\left(\tilde{k}\right)$. Let l be a prime such that the char$\left(\tilde{k}\right)\ne l$. If the ${\text{Br}}_l\text{dim}\left(\tilde{k}\right)=d$, then we also show that ${\text{Br}}_l\text{dim}\left(k\right)\le d+n$. Let C be a curve over k and $F=k\left(C\right)$. Then we show that any quadratic form over F with dimension $>2^{n+1}{u}_s\left(\tilde{k}\right)$ is isotropic over F. We further show that if ${\text{Br}}_l\text{dim}\left(\tilde{k}\right)\le d$ and ${\text{Br}}_l\text{dim}\left(\tilde{k}\right(T\left)\right)\le d+1$, then ${\text{Br}}_l\text{dim}\left(F\right)\le d+n+1$.