Why the nth-root function is not a rational function
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David E. Dobbs [1]
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[1] University of Tennessee
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- Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 48, Nº. 7, 2017, págs. 1120-1132
- Idioma: inglés
- DOI: 10.1080/0020739x.2017.1319980
- Texto completo no disponible (Saber más ...)
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Resumen
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.
