**ERIC Number:**EJ1152147

**Record Type:**Journal

**Publication Date:**2017

**Pages:**13

**Abstractor:**As Provided

**ISBN:**N/A

**ISSN:**ISSN-0020-739X

**EISSN:**N/A

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.

Descriptors: Mathematics Instruction, Mathematical Concepts, Mathematical Logic, Calculus, Algebra, Validity

Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals

**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