  The aim of this article is to prove some results on the existence of an
integral extension domain of a complete local Noetherian domain in mixed
characteristic $p>0$ having certain distinguished properties with respect to
the Frobenius map. We prove the main results by constructing required extension
domains via Witt vectors and the method of maximal \'etale extensions. It is
worth remarking that the resulting algebras have deep connections with the
homological conjectures and the rings in $p$-adic Hodge theory.
