How do I go about expressing a group as a product of cyclic groups? For example, express:
$$O^*_K = \{\pm (24 + 5\sqrt{23})^r : r\in \mathbb{Z} \}$$
as a product of cyclic groups ($O^*_K$ is the group of units in $O_K = \mathbb{Z} [\sqrt{23} ]$). I was given that the answer is: $$ \mathbb{Z}/2 \times \mathbb{Z} $$
Could you please explain this in a general way so that I can do the same for any group:
$$O^*_K = \{\pm (a + b\sqrt{d})^r : r\in \mathbb{Z} \} \quad d>0$$
Thank you.