Resumen de Subsets of fields whose nth-root functions are rational functions

David E. Dobbs

• Let R be an integral domain with quotient field F, let S be a non-empty subset of R and let n 2 be an integer. If there exists a rational function ϕ: S → F such that ϕ(a) n = a for all a S, then S is finite. As a consequence, if F is an ordered field (for instance, R) and S is an open interval in F, no such rational function ϕ exists. Applications to finite fields and additional examples are given. The methods used are algebraic. A closing remark indicates how this note could be used as enrichment material in courses ranging from precalculus to undergraduate courses on abstract algebra or analysis.