# Math Help - Algebraic Numbers Proof

1. ## Algebraic Numbers Proof

A number r is algebraic if there is a nonzero polynomial p(x) with integer coefficients such that r is a root of p(x) = 0.

Prove that every kth root of a rational number r is an algebraic number.

2. ## Re: Algebraic Numbers Proof

Let $r$ be the k'th root of a rational number $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not 0. (i.e. $r^k = \frac{a}{b}$)

We know that $p(x) = x^k - \frac{a}{b}$ is a polynomial with a root at $r$

To ensure the coefficients are all integer we simply multiply the right hand side by $b$