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)
Theorem 5.5 Gauss's Lemma
If and f can be factored into a product of non-scalar polynomials in , then f can be factored into a product of non-scalar polynomials in ; each factor in is an associate of the corresponding factor in
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 can be factored non-trivially in , we need only check to see if it can be factored non-trivially in . The latter is presumably an easier task."
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 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 then it can be factored in ! 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 then it cannot be factored non-trivially in ???
Can anyone clarify this matter for me? Would appreciate the help.