Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say about the minimal possible $N$ here? 



Which power of $2$ kills $W(k)$?
You are asking about the exponent of the additive group of the Witt group of $k$.
The answer is two times the level of $k$.


It suffices to show for the forms of dimension one.
If $s$ is the level of $k$, i.e., $s=min\{n\ :-1=x\_1^2+\cdots+x\_n^2, x\_i\in k\}$ then every element of $W(k)$ is killed by $2s$. In fact as $s\times\langle1\rangle$ is anisotropic, the order of $\langle1\rangle$ is exactly $2s$. Also with same argument for every $a\in k^\times$, the order of $\langle a\rangle$ is at most $2s$.


This result is due to Albrecht Pfister
(Quadratische Formen in beliebigen Körpern.
Invent. Math. 1, 1966 116–132).


