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Demazure, lectures on p-divisible groups, IV.1, isogenies

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• Michel Demazure, lectures on p-divisible groups, web

Unless otherwise stated let $k$ be a perfect field of prime characteristic.

We denote write $B(K):=\mathrm{Quot}(W(k))$ for the quotient field of the Witt ring $W(k)$. We extend the Frobenius morphism $x\mapsto x^{(p)}$ to an automorphism of $B(k)$. The set of fixed points of $x\mapsto x^{(p)}$ in $W(k)$ is $W(F_p)={\mathbb{Z}}_p$. The set of fixed points of $x\mapsto x^{(p)}$ in $B(k)$ is $B(F_p)={\mathbb{Q}}_p$.