Determine $\mathscr{O}_K$ for the number field $K=\Bbb Q(\sqrt[3]{20})$. Show that $\mathscr{O}_K$ is not monogenic.
Let $\alpha=\sqrt[3]{20}$. We have $\Delta(\Bbb Z[\alpha])=-3^3(-20)^2=-2^4 3^3 5^2$.
By the Kummer-Dedekind Theorem that $\Bbb Z[\alpha]$ is regular above 3 and 5. Modulo 2, $f$ factors as $X^3$. We have $f=X\cdot X^2-20$, so the remainder of $f$ upon division by $X$, which is $20$, is divisible by $2^2$. The unique prime $\frak p_2$ above 2 is singular and we find an element $\beta=\frac{1}{2}\alpha^2\in \mathscr{O}_K\setminus \Bbb Z[\alpha]$ that satisfies $\beta^3=50$. Consider the integral extension $R:=\Bbb Z[\alpha,\beta]\supset \Bbb Z[\alpha]$ of index $2$. The discriminant of $R$ is $2^{-2}\Delta(\Bbb Z[\alpha])=-2^2 3^3 5^2$. Since $2$ divides both the index and the new discriminant, we have to check whether $R$ is regular above $2$.
My syllabus suggests the following procedure in a similar example but leaves out all the details. To check whether $R$ is regular above $2$, we check whether there are integral elements in $\frac12R \setminus R$. In order to do this, they find representatives for the finite factor group $\frac12 R/R$ and check if their lift is integral.
My question is: Could someone explain why this procedure works? And how would I go about finding representatives for $\frac12R \setminus R$? For example, I see that $[\frac12 \alpha]$, $[\frac12 \beta]$ are examples of classes and for example $[\frac12 \alpha^2]=[\beta]=[0]$.
