I am reading Anderson and Feil - A First Course in Abstract Algebra - Section 5.3 Polynomials with Integer Co-effficients

On page 60 A&F state Gauss's Lemma in the following way (see attachment)

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Theorem 5.5 Gauss's Lemma

If $\displaystyle f \in \mathbb{Z} [x] $ and f can be factored into a product of non-scalar polynomials in $\displaystyle \mathbb{Q} [x] $, then f can be factored into a product of non-scalar polynomials in $\displaystyle \mathbb{Z} [x] $ ; each factor in $\displaystyle \mathbb{Z} [x] $ is an associate of the corresponding factor in $\displaystyle \mathbb{Q} [x] $

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Then at the bottom of page 60 A&F state the following:

"We can rephrase Gauss's Lemma in the following way: To see whether a polynomial in $\displaystyle \mathbb{Z} [x] $ can be factored non-trivially in $\displaystyle \mathbb{Q} [x] $, we need only check to see if it can be factored non-trivially in $\displaystyle \mathbb{Z} [x] $. The latter is presumably an easier task."

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My problem is that at first glance this does not seem like a restatement of Gauss's Lemma - if we check and find that f can be factored non-trivially in $\displaystyle \mathbb{Z} [x] $ then we can conclude nothing since the implication of Gauss's Lemma goes the other way i.e. it says if f can be factored in $\displaystyle \mathbb{Q} [x] $ then it can be factored in $\displaystyle \mathbb{Z} [x] $! Problem!

UNLESS what A&F are referring to is the contrapositive of Gauss's Lemma - if you find that f cannot be factored non-trivially in $\displaystyle \mathbb{Z} [x] $ then it cannot be factored non-trivially in $\displaystyle \mathbb{Q} [x] $ ???

Can anyone clarify this matter for me? Would appreciate the help.

Peter