Let R be an integral domain with quotient field F and let
p (X) be a monic polynomial in R[X] : Assume that p (X) = a (X) b (X)
where a (X) and b (X) are monic polynomials in F [X] of smaller degree
than p (X) : Prove that if a (X)is not in R[X] then R is not a UFD(unique factorization domain). Deduce that Z[2sqrt2] is not a UFD.
R is not a UFD since Gauss' lemma does not hold; that is, if R is a UFD with field of fractions F, then every polynomial which is reducible in F[x] ...
This is a proof regarding a unique factorization domain.