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## Main Question or Discussion Point

Ok, the book I'm reading states Gauss's lemma as such:

If f(x) is a monic polynomial with integral coefficients that factors into two monic polynomials with coefficients that are rational, f(x) = g(x)h(x), then [tex]g(x), h(x) \in \mathbb{Z}[x][/tex].

Now one of the exercises says to prove that:

If f(x) is a polynomial with integral coefficients that factors into two polynomials with coefficients that are rational, f(x) = g(x)h(x), then there is a factoring [tex]p(x), q(x) \in \mathbb{Z}[x][/tex] such that f(x) = p(x)q(x).

Which seem like almost the same problem except the exercise is more general in the case that they do not have to be monic. Yet, with g(x) and h(x) having rational coefficients, can't they factored into polynomials from [tex]\mathbb{Z}[x][/tex] and, moreover, monic by pulling out a rational factor?

I guess what I'm asking is thru what means can you go from Gauss's lemma to the exercise in the simplest way...

If f(x) is a monic polynomial with integral coefficients that factors into two monic polynomials with coefficients that are rational, f(x) = g(x)h(x), then [tex]g(x), h(x) \in \mathbb{Z}[x][/tex].

Now one of the exercises says to prove that:

If f(x) is a polynomial with integral coefficients that factors into two polynomials with coefficients that are rational, f(x) = g(x)h(x), then there is a factoring [tex]p(x), q(x) \in \mathbb{Z}[x][/tex] such that f(x) = p(x)q(x).

Which seem like almost the same problem except the exercise is more general in the case that they do not have to be monic. Yet, with g(x) and h(x) having rational coefficients, can't they factored into polynomials from [tex]\mathbb{Z}[x][/tex] and, moreover, monic by pulling out a rational factor?

I guess what I'm asking is thru what means can you go from Gauss's lemma to the exercise in the simplest way...