2

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?

thehardyreader
  • 1,134

1 Answers1

2

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 $D=\mathbb Z$.

lhf
  • 216,483
  • Thanks for your answer! In my algebra course we introduced the quotient field of an integral domain $R$ via an equivalence relation $\equiv$ so $K$ would be $K=R\times R^\times / \equiv$. So the elements in the quotient field are equivalence classes and the coefficients of the polynomial are elements in $R$. Why can I mix multiply them? – thehardyreader Jun 11 '18 at 21:01
  • 1
    @thehardyreader, there is a natural embedding of $D$ into $K$, and so we may assume $D \subseteq K$. Focus not on what things are but rather on how they behave. – lhf Jun 11 '18 at 21:02
  • ah okay! yes that is clear to me. so I should see the coefficients as elements in $K$ rather then as elements of $R$ (or $D$ if you prefer). – thehardyreader Jun 11 '18 at 21:07
  • @thehardyreader, if it makes you feel better, write "Let $K$ be a field containing $D$" instead. – lhf Jun 11 '18 at 21:08
  • okay, thanks for your help! – thehardyreader Jun 11 '18 at 21:16