- Thread starter
- #1

- Jun 22, 2012

- 2,918

Prove that if f(x) and g(x) are polynomials with rational co-efficients whose product f(x)g(x) has integer co-efficients, then the product of any co-efficient of g(x) with any coefficient of f(x) is an integer.

My initial thoughts on this are that the exercise seems to be set up for an application of Gauss Lemma since we have that Z is a UFD with field of fractions Q and further we have \(\displaystyle p(x) \in Z[x] \) where p(x) = f(x)g(x) and \(\displaystyle f(x), g(x) \in Q[x] \).

Thus we apply Gauss Lemma (see attached) so

p(x) = (rf(x))(sg(x))

where \(\displaystyle rf(x), sg(x) \in Z[x] \)

But .... where to from here ... can someone please help me advance from here ...

Peter

[This problem has also been posted on MHF]

My initial thoughts on this are that the exercise seems to be set up for an application of Gauss Lemma since we have that Z is a UFD with field of fractions Q and further we have \(\displaystyle p(x) \in Z[x] \) where p(x) = f(x)g(x) and \(\displaystyle f(x), g(x) \in Q[x] \).

Thus we apply Gauss Lemma (see attached) so

p(x) = (rf(x))(sg(x))

where \(\displaystyle rf(x), sg(x) \in Z[x] \)

But .... where to from here ... can someone please help me advance from here ...

Peter

[This problem has also been posted on MHF]

Last edited: