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Let k denote an arbitrary field and let R be an affine local domain over k. Let (Q.k(R), <5£ ) be the universal algebra of ¿-higher differentials over R. Let K be the quotient field of R and L the residue class field of R. If A: is a separable extension of k and £ is a separable algebraic extension of k, then it is shown that R is a regular local ring if and only if Qk(R) is a free Ä-algebra. If both K and L are separable extensions of k and R has a separating residue class field, then R is adoi:10.1090/s0002-9939-1972-0300999-0 fatcat:duoh2lckvnfhffo2k6nrdy44p4