Idempotents in the endomorphism ring of an ideal in a $p$-extension of a complete discrete valuation field with residue field of characteristic $p$ as a Galois module

In this paper we study the question when there exist non-trivial idempotents in the endomorphism ring of an ideal in a $p$-extension of a complete discrete valuation field with residue field of finite characteristic $p>2$ as a Galois module. We prove that there are no non-trivial central idempotents for a non-abelian totally widely ramified extension.