Rational root theorem

Proposition

Let \(R\) be an UFD, \(f = \sum_{i=0}^{n} \alpha_i X^i \in R[X]\) a polynomial.
Suppose \(p,q\) are coprime elements, such that in the quotient ring

\begin{align*} f(\frac{p}{q}) = 0 \end{align*}

Then it follows, that

\begin{align*} p \mid& \alpha_0 \\ q \mid& \alpha_n \end{align*}

Proof

a)

We get

\begin{align*} 0 =& f(\frac{p}{q}) \\ 0 =& \sum_{i=0}^{n} \alpha_i \left(\frac{p}{q}\right)^i && \vert \cdot q^n 0 =& \alpha_n p^n + \alpha_{n-1} p^{n-1} q ... + \alpha_{1} p q^{n-1} + \alpha_0 q^{n} \\ - \alpha_0 q^n =& p \cdot (\alpha_n p^{n-1} + ... + \alpha_1 q^{n-1} \end{align*}

Hence

\begin{align*} p \mid -\alpha_n q^n \end{align*}

and since \(p,q\) are coprime, we conclude that

\begin{align*} p \mid -\alpha_0, \alpha_0 \end{align*}

b)

\begin{align*} 0 =& f(\frac{p}{q}) \\ 0 =& \sum_{i=0}^{n} \alpha_i \left(\frac{p}{q}\right)^i && \vert \cdot q^n 0 =& \alpha_n p^n + \alpha_{n-1} p^{n-1} q ... + \alpha_{1} p q^{n-1} + \alpha_0 q^{n} \\ - \alpha_n p^n =& \alpha_{n-1} p^{n-1} q ... + \alpha_{1} p q^{n-1} + \alpha_0 q^{n} \\ - \alpha_n p^n =& q \cdot (\alpha_{n-1} p^{n-1} + \alpha_{n-2} p^{n-2} q ... + \alpha_0 q^{n-1}) \end{align*}

Hence

\begin{align*} q \mid -\alpha_n q^n \end{align*}

and since \(p,q\) are coprime, we conclude that

\begin{align*} p \mid -\alpha_n, \alpha_n \end{align*}