This may be a somewhat dumb question, but I have recently learned of Gauss's lemma, from 2 different books, that give 2 different formulations and I must be having some kind of mental block, because I can't figure out how they are equivalent.
The first formulation is:
Let $f$ in $\Bbb Z[x]$ be the product of two monic (or equivalently of content $0$, if I understand this correctly) polynomials $g$ and $h$ in $\Bbb Q[x]$. Then $g$ and $h$ themselves are members of $\Bbb Z[x]$.
The second formulation is:
Let $f$ in $\Bbb Z[x]$ be irreducible over $\Bbb Z$. Then $f$, considered as a member of $\Bbb Q[x]$, is also irreducible over $\Bbb Q$.
Can someone please point out to me how exactly these two statements are equivalent? I feel like they're contrapositives of each other, but straight up negating and flipping either doesn't produce the other...
Edit: So just to clarify what I'm having trouble with, the contrapositive of the second statement should, in my opinion, be "If $f$ is reducible over $\Bbb Q$, then it's reducible over $\Bbb Z$". But who's to say that the factors of $f$ over $\Bbb Z$ will be the same as over $\Bbb Q$?