Certain properties of traces on a finite-dimensional associative algebra A lead to the definition of an element t(A) e H'(Out A, C*), C* being the multiplicative group of the center of A as Out A-module. It is shown that t(A)=O is equivalent to the existence of nondegenerate traces on A which are invariant under composition with all automorphisms of A. In particular, by means of Galois theory, t(A)=O is shown for a semisimple algebra A, whereas t(A)00 for certain group algebras. 1. Let R be a field, A an associative unitary algebra of finite dimension over R. By a trace on A we mean a linear map r: A--?R such that -r(ab)=7-(ba) Va, b E A. This is one possible generalization of the notion of a trace on matrix rings (see [4]; for a generalization in another context, see [2]). In ??2-4 we shall list some generalities on traces; let T(A) be the R-vectorspace of all traces on A. 2. The existence of nonzero traces on A depends on the abelianized algebra Aa. Let [A, A] be the vectorspace generated by all commutators [a, b]=ab-ba in A, Aa the quotient A/[A, A]. The class map n-:A--*Aa provides an isomorphism of vectorspaces 7r*:HomR(Aa, R) -T(A), where v* is the dual map of vz. One knows that Aa$(O) if A is simple [1], hence (2.1) T(A) $ (0) for a simple algebra A. 3. The radical of a trace -r is the set Rr = {a E A/r(ab) = 0 Vb E A}, Received by the editors February 12, 1973 and, in revised form, May 17, 1973. AMS (MOS) subject classifications (1970). Primary 16A49, 16A40; Secondary 16A26, 18A10.