Buhler gave a criterion for a class of finite free extensions of discrete valuation rings to be tamely ramified 1-dimensional regular rings. In this note, we extend this criterion to finite free extensions of general local rings and, in the final section, indicate the extension to schemes. THEOREM 1. If A is regular (resp. gr /ί (m /4 ) has no zero divisors) and if B = A[X]/(f(X)) where f(X) is a monicpolynomial and κ(m) -> κ(n,) is separable for all i = 1,... ,s 9 then PB/A) ^ Σ K,/m, " l)[κ(n

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... z ): κ(m A )] i=\ with equality if and only //(resp. only if) ed(2?) = ed(A) and B is tamely ramified over A in that p \ e n /m for all /, where p is the characteristic of κ{xn A ). 71 72 M. L. BROWN