$f$ has a root in $k \implies$ that $f$ is reducible...only true for $k = $ field?

Question

Let $f \in k[x]$, where $k[x]$ is a ring of polynomials with coefficients in $k$. Suppose: $f$ has a root in $k \implies$ that $f$ is reducible. Is this statement only true when $k = $ a field?

Answer 1

Let $R$ be a (commutative) ring. If $f\in R[X]$ has a root then $f$ has a linear factor:

polynomial with a root but no linear factor

This does not always imply that $f$ is reducible, consider for example $f=2X\in \Bbb Q[X]$ which has a rational root but is irreducible.

However, for non-constant polynomials, or monic polynomials over an integral domain $R$, having a root implies reducibility.

Definition: A polynomial $f\in R[X]$ over an integral domain $R$ is irreducible if $f$ is not a unit in $R$ and whenever $f=gh$ with $g,h\in R[X]$, we have $g$ or $h$ is a unit of $R$.