# Cohomological dimension of transcendental p-adic extensions

Let $q$ denote a quadratic form over a field $k$.

The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q))$ | $q$ is anisotropic over $k\}$.

Let $k = \mathbb{Q}_p$ for any prime $p$ and set

$L = k(t_1,..,t_n)$.

It is known that $u(k)=4$ and newer results by David B. Leep state that

$u(L) = 4\cdot2^n = 2^{n+2}$.

As a consequence from the Arason-Pfister Hauptsatz we have that

$2^{cd(L)} \leq u(L) = 2^{n+2}$, while $cd(L)$ denotes the cohomological dimension of $L$.

Is $cd(L) = n+2$ i.e. does equality hold in the upper equation?

This question is not trivial in general as Serre points out by mentioning results of Merkurjev in Galois Cohomology. Merkurjev constructs fields $k$ with $cd(k)=2$ having any desired even $u(k) \geq 2$.

• Can you say what $u$ is? Commented Oct 23, 2014 at 0:28
• Its the u-invariant of k. I will include its definition in my first post.
– nxir
Commented Oct 23, 2014 at 0:37

I think the answer to your question can be found in Serre's "Galois cohomology" book, Section II.4.2, where a more general result is proved for transcendental field extensions.

• Thanks! I cant believe i overlooked it. Edit: And the answer is "Yes"!
– nxir
Commented Apr 20, 2015 at 23:08