Concerns with the proof of the rational root theorem

Question

I have to solve the following exercise:

Let $R$ be a unique factorization domain and $f \in R[x]$ a polynomial with leading coefficient $a_n$ and constant coefficient $a_0$. Furthermore $p,q \in R$ coprime with $f(\frac{p}{q})=0$. Prove $p\vert a_0$ and $q\vert a_n$.

I have seen the proofs on wikipedia https://en.wikipedia.org/wiki/Rational_root_theorem) and in this thread Rational root theorem.

I understand the arguments but my concern is am I allowed to multiply with $q^n$ and shorten the fractions in an UFD? Especially what means $ \frac{p}{q}$ in a UFD? My guess was that this is an element of the quotientfield in it's unique reduced representation. But isn't this mixing of elements of different types then?

Answer 1

The rational root theorem for UFDs is this:

Let $D$ be a UFD. Let $K$ be its field of fractions. Let $p/q \in K$, with $p,q \in D$ coprime, be a solution of a polynomial equation over $D$: $$a_n z^n + a_{n-1}z^{n-1} + \cdots + a_1 z + a_0 = 0$$ Then $q$ must divide $a_n$ and $p$ must divide $a_0$.

The proof is the same as for $D=\mathbb Z$.