Formal functions on prevalued domains

For a nonzero, univariate power-series f with coefficients in an integral domain R, we prove some results about the cardinality of certain zero-sets of f. We prove formal analogues of Hilbertʼs Theorem 90 for R〚X1,…,Xn〛 which generalize the known results. Given a torsion group G of R-automorphisms (resp. continuous R-automorphisms) of R〚X1,…,Xn〛, we show that if R contains Q, then G is isomorphic to a subgroup of GL(n,R/I) (resp. GL(n,R)), where I denotes the Eakin-Sathaye ideal of R.