Why the n^th-root function is not a rational function

ABSTRACT

The set of functions $\{x^q\mid q\in \mathbb{R}\}$ is linearly independent over $\mathbb{R}$ (with respect to any open subinterval of (0, ∞)). The titular result is a corollary for any integer n ≥ 2 (and the domain [0, ∞)). Some more accessible proofs of that result are also given. Let F be a finite field of characteristic p and cardinality p^k. Then the p^th-root function F → F is a polynomial function of degree at most p^k − 2 if p^k ≠ 2 (resp., the identity function if p^k = 2). Also, for any integer n ≥ 2, every element of F has an n^th root in F if and only if, for each prime number q dividing n, q is not a factor of p^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.

KEYWORDS:

• n^th root
• rational function
• linear function
• linear independence
• finite field
• characteristic
• polynomial
• degree
• primitive root
• Lagrange's theorem