TITLE: What are the units of Z[x]?
QUESTION [6 upvotes]: Where $\mathbb{Z}[x]$ is the ring of polynomials in $x$ with integer coefficients. The book I am studying says the unity of this ring is $f(x) = 1$ so then if some $p \in \mathbb{Z}[x]$ is a unit, this then must mean that $p^{-1} \in \mathbb{Z}[x]$, where $p \cdot p^{-1} = 1$, correct? 
Obviously then $f(x) = 1$ and $f(x) = -1$ are units, and I have seen elsewhere on the internet that people claim these are the only units of $\mathbb{Z}[x]$, but aren't simple one-term polynomials also units? That is, $\forall k \in \mathbb{Z}, f(x) = x^k$ is a unit because $(f(x) = x^k)^{-1} \Longleftrightarrow f(x) = x^{-k}$ and $x^k \cdot x^{-k} = 1$, no?

REPLY [2 votes]: More generally, the units of $D[x]$ are exactly the units of $D$, when $D$ is a domain.