odd degree $0$-cycles and rational points on a quadric hypersurface

Question

Is it true that a smooth quadric hypersurface has a rational point if and only if it has an odd degree $0$-cycle?

I think this is true. If so, can someone give a (geometric) proof?

Answer 1

Yes. It was a conjecture of Witt, proved by Springer, that a quadric hypersurface, over a field of characteristic not equal to $2$, has a rational point if and only if it has a point over a field extension of odd degree. This was proved in:

Springer, Tonny Albert - Sur les formes quadratiques d'indice zéro. (French) C. R. Acad. Sci. Paris 234, (1952). 1517–1519.

It is not too difficult to see that this implies the result you want, when the field does not have characteristic $2$, see for instance:

Swan, Richard G. - Zero cycles on quadric hypersurfaces. Proc. Amer. Math. Soc. 107 (1989), no. 1, 43–46.

I don't know what happens over fields of characteristic $2$.