## Deduce the rational root theorem from Gauss’ Lemma

Use Gauss’ lemma to prove the Rational Root Theorem: If $R$ is a Euclidean domain with fraction field $Q$ and if $p(x) = \sum_{i=0}^n c+i x^i$ is a polynomial in $R[x]$, then any root of $p(x)$ in $Q$ has the form $\frac{a}{b}$ where $a|c_0$ and $b|c_n$.

Suppose $\frac{a}{b}$ is a root of $p(x)$ in $Q$ in lowest terms (i.e. $a$ and $b$ are relatively prime). Now $0 = \sum_{i=0}^n c_i \frac{a^i}{b^i}$, so that (multiplying both sides by $b^n$) $0 = \sum_{i=0}^n c_i a^i b^{n-i}$.

Now $-c_0 b^n = \sum_{i=1}^n c_ia^ib^{n-i}$. Since $a$ divides each term on the right hand side, $-c_0 b^n = at$ for some $t$. Since $a$ and $b$ are relatively prime, $a$ and $b^n$ are relatively prime. Thus $a|c_0$, by a previous exercise.

Similarly, $-c_na^n = bs$ for some $s$, and we have $b|c_n$.