I'm trying to determine $U(\mathbb{R}[x])$, where $U(R)$ denotes the unit group of a ring $R$.
I think the answer is all non-zero constant polynomials, but I'm having trouble showing that these are the only units.
I'm trying to determine $U(\mathbb{R}[x])$, where $U(R)$ denotes the unit group of a ring $R$.
I think the answer is all non-zero constant polynomials, but I'm having trouble showing that these are the only units.
If you look at the degree, if $PQ=1$ then $\deg(PQ)=\deg(P)+\deg(Q)=0$ (if $R$ is a domain) so $P$ and $Q$ are non-zero constant (because $\deg(P)=\deg(Q)=0$). And the constant has to be invertible.
And if $P(X)=a\in U(R)$, $P^{-1}(X)=a^{-1}$ is a multiplicative inverse.
So, if you are in a domain $U(R[X])=U(R)$.