Fraleigh says:
"It is worthwhile to remember that the units in $F[x]$ are precisely the nonzero elements of F. Thus we could have defined an irreducible polynomial $f(x)$ as a nonconstant polynomial such that in any factorization $f(x) = g(x)h(x)$ in $F[x]$, either $g(x)$ or $h(x)$ is a unit."
I'm quite confused about this. I think his first sentence is saying that because $F$ is a field, every element is either $0$ or a unit. But I don't see how the second sentence follows at all. Just let $g(x)$ and $h(x)$ be two polynomials with degree lower than $f(x)$, and $g(x)h(x) = f(x)$. Aren't $g(x)$ and $h(x)$ automatically units since they are in $F[x]$ and $f(x)$ is nonconstant (and hence nonzero)? So I don't see how restricting $g(x)$ or $h(x)$ to be a unit is a restriction at all, or shows that we don't have $g(x)h(x) = f(x)$.
Every nonzero element of $F$ is a unit. But not every element of $F[x]$ is a unit.
For instance every nonzero real number is a unit in $\mathbb R$, but nonconstant polynomials are not units in ${\mathbb R}[x]$.