Exercise on polynomial and GCD

Question

Let $f(x), g(x)$ be relatively prime polynomial with coefficients in $\mathbb{Z}$. How can I prove that the GCD $(f(n),g(n)) = O(1)$ for $n \to \infty$, $n \in \mathbb{N}$? Thank you for the help!!

Answer 1

In general, if $A$ is a so-called UFD (unique factorisation domain), then the one indeterminate polynomial ring $A[X]$ will have the same property, so in particular it will be a GCD-ring (i.e. such that any two elements admit a greatest common divisor). If $F$ denotes the fraction field of $A$ and $f, g \in A[X]$ two polynomials, then any greatest common divisor $h$ they have over $A$, it will remain a g.c.d for the pair over $F$; on the other hand, as $F[X]$ is a PID (principal ideal domain), the g.c.d can be expressed as $h=fk+gl$ for $k, l \in F[X]$. In your particular case, it must be that $1=fk+kl$ for $k,l \in \mathbb{Q}[X]$. Clearing denominators, this relation can be brought to the form: $d=fk+gl$, for (a renewed pair of polynomials) $k, l \in \mathbb{Z}[X]$ and $d \in \mathbb{Z}$. From here you can conclude that $(f(n), g(n))\mid d$, for any $n \in \mathbb{Z}$. Now, can you see how this answers your question?

Answer 2

Here's an outline:

• Use Gauss's Lemma to show that polynomials that are relatively prime over $\mathbb{Z}$ remain relatively prime over $\mathbb{Q}$.

• Let $p(x)$ and $q(x)$ in $\mathbb{Q}[x]$ be such that $p(x)f(x) + q(x)g(x) = 1$.

• Clear denominators to get an equation in $\mathbb{Z}[x]$.

• The conclusion follows.