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Two exact cosmological solutions to a scalar-field potential motivated by six-dimensional (6D) Einstein-Maxwell theory are given. The resulting pure scalar-field cosmology is free of singularity and causality problems but conserves entropy. These solutions are then extended into exact cosmological solutions for a decaying scalar field with an approximate two-loop 4D string potential. The resulting cosmology is, for both solutions, free of cosmological problems and close to the standard cosmology of the radiation era.PACS numberh): 98.80.Cq, 98.80.Dr Scalar-field cosmology has received, and is currently receiving, much attention. One usually assumes a scalar field with Lagrangian densitywhere R is the curvature scalar. A particular model is specified by a suitable choice of the potential. The field of choice is rather wide, ranging from Coleman-Weinberg 111 and exponential potentials [2] to k$4 or a simple rn 2$2 term 131. Scalar-field potentials which are formed by different terms in different intervals of the range of $ have also been considered 141. The conclusions of these scalar-field cosmological models are naturally strongly model dependent. There are, however, attempts to base the choice of the potential on some fundamental dynamical considerations. Prominent among these are field potentials that are, under certain approximations, deduced from higher-dimensional supergravity 151 and superstring [61 theories. These are invariably of the exponential type. In particular, we quote the potential the degree of freedom of the radius of the internal sphere.-In higher-dimensional superstring theories the scalar field is among the matter fields that contribute to the action and effective potential of the theory. Loop expansion [61, or expansion in the number of interacting particles [8l, of the action leads to a perturbative expression of the 4D potential of the form where the coefficients a, are functions of the nonscalar fields.An exact solution with a cosmological scale factor has been obtained 191 for the one-loop potential. It has also been proved 191 that all expanding solutions of the one-loop potential approach solution (4) for large t.In this paper we first report on two exact solutions for the two-term potential where K =m, obtained [7] from a four-dimensional in which the nonscalar fields are neglected on the assump-(4D) action based on a 6D Einstein-Maxwell theory in tion that they are slowly varying during the period of which the gauge vector field assumes a monopole con-scalar-field dominance. One notes that the exponential figuration on s2. In this case the scalar field arises from factor in ( 5 ) is similar to (2) which differs by & from 45 R997