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 $k$.
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1$\begingroup$ I suppose that you meant the Witt ring, and the rank modulo 2 ? $\endgroup$– GreginGreNov 17, 2022 at 16:19
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$\begingroup$ @GreginGre yes. $\endgroup$– Nanjun YangNov 17, 2022 at 22:07
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1 Answer
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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.
