Degree \(p\) extensions of an unramified regular local ring of mixed characteristic \(p\) (Q1081647)

Let \(R\) be an unramified regular local ring of mixed characteristic \(p\). The ``direct summand conjecture'', for module-finite extension rings of a regular Noetherian ring, can be reduced to the case of such a ring \(R\) [see \textit{M. Hochster}, J. Algebra 84, 503--553 (1983; Zbl 0562.13012)]. Let \(K\) be the fraction field of \(R\) and let \(u\in R\). It is shown that, for \(L=K(\root p\of{u})\) (and, also, for certain degree \(p^ 2\) extensions \(L\)), \(R\) is a direct summand of the integral closure of \(R\) in \(L\). A related example is given of a normal but non-Cohen-Macaulay ring.