0. Introduction. We shall prove that the ring of continuous complex-valued functions on a completely regular Hausdorff space is totally integrally closed iff the space is extremally disconnected. X always denotes a completely regular Hausdorff space. R is the field of real numbers, C of complex numbers. C(X) denotes the ring of all continuous R-valued functions on X, C*(X) the subring of bounded functions in C(X). i = a/(-1). Then C(X) [i] (respectively, C*(X) [i]) is the ring of continuous (respectively, bounded continuous) C-valued functions of X. A ring A (ring commutative ring with 1) is totally integrally closed if for every integral extension h: B-*C, the induced map h*: Hom(C, A) -*Hom(B, A) is surjective. See [1] and [4] for a detailed treatment. By [4, Theorem 1 and Proposition 5], A is totally integrally closed iff the following four conditions hold: (1) A is reduced, (2) every monic polynomial of positive degree in one indeterminate over A factors completely into monic linear factors, (3) distinct minimal primes of A are comaximal, and (4) the set of minimal primes of A, Min Spec A, is an extremally disconnected space in the inherited Zariski topology.