Which field extensions of $\mathbb Z$ are UFD?

Question

As we know, $\mathbb Q$ is a field and $\mathbb Z$ is the ring of integers in it, which is an UFD. Also let $E$ is an extension of $\mathbb Q$. Then which type extensions of $\mathbb Q$ guaranteed that the set of integers of $E$ is an UFD. I didn't even know where to start. I got this curiosity when I studied $\mathbb Z[i]$.

Answer 1

When $E$ is a number field, that is, a finite degree extension of $\mathbb Q$, and $\mathcal O_E$ its ring of integers, then the following are equivalent:

• $\mathcal O_E$ is a UFD;
• $\mathcal O_E$ is a PID;
• The class number $h_E = 1$.

For more on what is known and what is conjectured, see Class number problem on Wikipedia.