ON EXTENSIONS OF SEMILOCAL PRUFER DOMAINS

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prufer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prufer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prufer domain. Moreover, we show that any two extensions of a semilocal Prufer domain are incomparable (by inclusion) in an algebraic extension of fields.