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Let $p$ be a rational prime and $K$ a complete discrete valuation field with residue field $k$ of positive characteristic $p$. When $k$ is finite, generalizing the theory of Deligne [], we construct in [] and [] a theory of local ${\epsilon }_0$-constants for representations, over a complete local ring with an algebraically closed residue field of characteristic $\ne p$, of the Weil group $W_K$ of $K$. In this paper, we generalize the results in [] and [] to the case where $k$ is an arbitrary perfect field.