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at the moment I am working on a few exercises concerning Witt rings but my knowledge on them is still very shallow. I would like to attain more insight in them so maybe you can help me with those questions:

Let $k$ be a field of positive characteristic. Denote by $W(k)$ the witt ring of $k$, by $V$ the Verschiebung map and by $F$ the Frobenius map.

a) For $a,b \in W(k)$ we have $V^m(a)\cdot V^n(b)=V^{m+n}(F^n(a)\cdot F^m(b))$

b) $k$ perfect iff $W(k)/ pW(k)$ reduced

c) $W(k)$ is a local integral domain with maximal ideal $im(V)$

d) $W(k)$ noetherian iff $k$ perfect

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If you would like to develop your understanding of the Witt ring deeply, then you should look to T.Y. Lam's text "Introduction to Quadratic Forms Over Fields" which gives a comprehensive examination of Witt rings of fields of characteristic other than 2.

Otherwise, you might be interested in the following link, which presents the Witt ring from a less traditional viewpoint, and addresses both maps of Frobenius and Verschiebung

http://www.math.harvard.edu/~rabinoff/misc/witt.pdf

If you find yourself even more curious about these objects, I recommend Marshall's text "Abstract Witt Rings." Which will present you with many odd open problems to ponder.