$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two Euclidean domains having exactly one invertible element. My question is:
Can we characterize all Euclidean domains $D$ having exactly one invertible element, i.e., $|D^{\times}|=1$ ?
For finite domains it is easy because any finite ED is an ID so a field, so the only one is $\mathbb Z_2$, but what about infinite Euclidean domains ? Please help. Thanks in advance
UPDATE : Obviously, as MooS has also noted, if $D$ is an ID with $|D^{\times}|=1$ then $D$ has characteristic $2$. Any progress regarding the problem is highly appreciated.