Why the nth-Root Function is Not a Rational Function

Autor/in	Dobbs, David E.
Titel	Why the nth-Root Function is Not a Rational Function
Quelle	In: International Journal of Mathematical Education in Science and Technology, 48 (2017) 7, S.1120-1132 (13 Seiten)Infoseite zur Zeitschrift PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0020-739X
DOI	10.1080/0020739X.2017.1319980
Schlagwörter	Mathematics Instruction; Mathematical Concepts; Mathematical Logic; Calculus; Algebra; Validity + Suchen Sie Ihr Suchwort? Mathematics lessons; Mathematikunterricht; Mathematical logics; Mathematische Logik; Analysis; Differenzialrechnung; Infinitesimalrechnung; Integralrechnung; Gültigkeit
Abstract	The set of functions {x[superscript q] \| q[element of][real numbers set]} is linearly independent over R (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 F be a finite field of characteristic p and cardinality p[superscript k]. Then the pth-root function F [right arrow] F is a polynomial function of degree at most p[superscript k] - 2 if p[superscript k] ? 2 (resp., the identity function if p[superscript k] = 2). Also, for any integer n = 2, every element of F has an nth root in F if and only if, for each prime number q dividing n, q is not a factor of p[superscript k] - 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. (As Provided).
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Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2020/1/01