Proofs of certain properties of irrational roots

TL;DR

This paper provides two elementary proofs, accessible to pre-calculus students, demonstrating that an irreducible irrational n-th root of a positive rational cannot satisfy a polynomial of degree less than n with rational coefficients.

Contribution

It offers simplified, accessible proofs of a classical algebraic property, making the concept understandable at a pre-calculus level.

Findings

Irreducible irrational n-th roots do not satisfy lower-degree rational polynomials.

Elementary proofs are provided suitable for pre-calculus students.

Several simple consequences of the main property are also proved.

Abstract

We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a polynomial of degree less than n with rational coefficients. We also state and prove a few simple consequences.