Resumen de Why the nth-root function is not a rational function

David E. Dobbs

• The set of functions is linearly independent over (with respect to any open subinterval of (0, 8)). The titular result is a corollary for any integer n= 2 (and the domain [0, 8)). Some more accessible proofs of that result are also given. Let Fbe a finite field of characteristic pand cardinality pk. Then the pth-root function F? Fis a polynomial function of degree at most pk- 2 if pk? 2 (resp., the identity function if pk= 2). Also, for any integer n= 2, every element of Fhas an nthroot in Fif and only if, for each prime number qdividing n, qis not a factor of pk- 1. Various parts of this note could find classroom use in courses at various levels, on precalculus, calculus or abstract algebra. A final section addresses educational benefits of such coverage and offers some recommendations to practitioners.