Prove that any polynomial in $F[x]$ can be written in a unique manner as a product of irreducible polynomials in $F[x]$.
$F$ is a field of fractions of a UFD.
Can someone tell me how I can solve this problem. It is one of my lecture note result, but my lecturer did not provide a proof.
I am assuming the letter $F$ was chosen to indicate a field, although the question does not say so.
This is just an instance of the general fact that Euclidean domains (or more generally principal ideal domains) are unique factorisation domains. The usual proof passes by establishing Euclid's lemma for irreducible elements, which serves to show that given two factorisations, any irreducible factor in the factorisation on the left can also be found (up to an invertible factor) in the factorisation on the right. The proof really matches that for the integers in all respects.