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ERIC Number: EJ1152147
Record Type: Journal
Publication Date: 2017
Pages: 13
Abstractor: As Provided
ISSN: ISSN-0020-739X
Why the nth-Root Function is Not a Rational Function
Dobbs, David E.
International Journal of Mathematical Education in Science and Technology, v48 n7 p1120-1132 2017
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.
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Publication Type: Journal Articles; Reports - Descriptive
Education Level: N/A
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A
Grant or Contract Numbers: N/A