Units of $\mathbb{Z}[\omega]$ where $\omega = \frac{1}{2} (1+ \sqrt{a})$, $a<0$ a square free integer.

Question

Wondering about this question, which is a sort of generalized form of the Eisenstein integers, let: $$ \omega = \frac{\sqrt{a}+1}{2} $$ Where $a \equiv 1 \textrm{ mod 4}$ and $a<0$. How does one show that the units of $\mathbb{Z}[\omega] = \{x + y \omega : x,y \in \mathbb{Z}\}$ are only $1$ and $-1$ unless $a=-3$? I have seen that this is not necessarily a euclidian domain for negative $a$. I have tried expanding out terms on the assumption that there is an inverse for something of the form $x+\omega y$, but this hardly simplified things, does anyone have a better argument to get the result?

Answer 1

If $\omega$ is a unit then $N(\omega)=\pm1$. This means that you have the equation $$x^2+ay^2=\pm 4.$$ In the case $a<0$ is is easy to show this has no solutions except for $a=-1,-3$.