Suppose $k$ is a field. I wonder when the Witt ring of the quadratic forms $\textbf{W}(k)$ has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a sufficient condition on $k$.



Projectivity of the fundamental ideal of Witt groups
You have two easy families of examples.


* $k$ is a quadratically closed field, that is a field in which every element is a square. In this case , $I(k)=0$, and is free.
* $k$ is a Euclidean field, that is a field in which squares form an ordering. Real closed fields are euclidean, but there exist euclidean fields which are not real closed. In this case, $I(k)$ is free, generated by $\langle 1,1\rangle$.


I convinced myself that these two cases are the only ones which can happen, but I will have to write up the proof to be sure.


