Definition of ring of dual numbers

In an exercise from Vakil’s algebraic geometry notes, he asks us to describe the set $rm Specspace k[epsilon]/(epsilon^2)$, where $k$ is a field. A comment from this question gives a solution, but it’s under the assumption that $epsilon$ is transcendental, and so the assumption may be made that $k[epsilon]$ is a PID. However, I am not sure why we can assume this. In the question, Vakil says "you should think of $epsilon$ as a very small number, so small that its square is 0 (although itself is not zero)". If $epsilon$ is any number we want, they we can certainly choose a value for which $k[epsilon]$ is not a PID. Am I missing something?