# Units in a quotient of a quadratic number ring

Let $$K$$ be a imaginary quadratic field. Let $$\mathcal O$$ be its ring of integers. Suppose that $$2$$ divides the discriminant of $$K$$.

What is the structure of the multiplicative group $$(\mathcal O /2^k \mathcal O)^\times$$?

• This is almost a duplicate of this question, asked at almost the same time: math.stackexchange.com/questions/3180079/… – Furlo Roth Apr 10 at 0:27
• And just as in that case, there is an isomorphism $\mathcal{O}^{\times}_2 \simeq \mu_{K_2} \oplus (\mathbf{Z}_2)^2$, where $\mu_{K_2}$ denotes the roots of unity in $K_2$, which (under your hypothesis) is either cyclic of order $2$ or of order $4$. Once again, the answer is best understood by learning about local fields. – Furlo Roth Apr 10 at 0:32
• Have you tried looking at a specific ring, like, say, $\mathbb Z[\sqrt{14}]$, and seeing if it gives you any general insights? – Robert Soupe Apr 10 at 4:07
• – nguyen quang do Apr 13 at 12:31