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I have a really basic question about the Witt Groethendieck ring of a field: In Lam's book, it says that $\widehat{W}(F)/\widehat{I}^2(F)$ depends only on the square classes of $F$, $\widehat{W}/\widehat{I}^2\cong \mathbb{Z}\oplus \widehat{I}/\widehat{I}^2$. We have a short exact sequence

$$0\to \widehat{I}/\widehat{I}^2\to \widehat{W}/\widehat{I}^2\to \widehat{W}/\widehat{I}\to 0$$

Why is this sequence split? What is the natural way to get a map from $\widehat{W}/\widehat{I}^2\to \widehat{I}/\widehat{I}^2$?

Chickenmancer
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user220463
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1 Answers1

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Lam indicates that this is split exact precisely when $-1=1\in \dot{F}/\dot{F}^2.$ See proposition 2.1, page 31 [Introduction to Quadratic Forms over Fields GSM volume 67.]

Chickenmancer
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