TITLE: No constant polynomials are irreducible?
QUESTION [5 upvotes]: No constant polynomials are irreducible?
What does it mean?
Is it because constant polynomials are units?

REPLY [8 votes]: You're correct, if working over a field; that is, if the polynomial ring you're talking about is $F[x]$ where $F$ is a field, then any non-zero constant polynomial is a unit of the ring $F[x]$, and the constant zero polynomial is the zero element of the ring $F[x]$, and therefore none of them are irreducible elements of $F[x]$, simply because the definition of "irreducible" excludes such elements.
However, if the polynomial ring is $R[x]$ where $R$ is a ring that is not a field, then there may be non-zero constant polynomials that are irreducible elements of $R[x]$. For example, in the ring $\mathbb{Z}[x]$, the element $2$ is a constant polynomial that is irreducible.