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In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have

$u(\mathbb{C}((t_1),..(t_n))) = 2^n$.

What do we get if we replace $\mathbb{C}$ with the p-adic numbers $\mathbb{Q}_p$, which have $u(\mathbb{Q}_p)=4$ ?

I think this question is unknown in general, are there any new results?

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    $\begingroup$ If $K$ is a $p$-adic field the u-invariant of $K(t)$ is now known to be $8$; more generally this holds for finite extensions of such fields. I think this was proved by Parimala and Suresh for $p$ odd and Heath-Brown for $p = 2$. $\endgroup$
    – naf
    Sep 20, 2014 at 6:46
  • $\begingroup$ Yes i found this paper of Parimala and Suresh. The proof involves symbol lengths of symbols in galois cohomology. It doesnt look at all like the general case could be derived easily from that. $\endgroup$
    – nxir
    Sep 20, 2014 at 8:28

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My question has been answered positiv by David B. Leep in his publication

The u-invariant of p-adic function fields - Journal für die reine und angewandte Mathematik, Band 2013, Heft 679 (Jun 2013).

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