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Let $\left\{ 1 , \omega_1, \dots , \omega_{n-1} \right\}$ be an integral basis for an order $\mathcal{O}$ of a number field of degree $n$ over $\mathbb{Q}$, let $\Delta $ be the discriminant of $\mathcal{O}$, and let $D \left( \omega_j \right)$ denote the discriminant of $\omega_j$. Is there a result that connects the discriminant of $\mathcal{O}$ to the discriminants of the generators of an integral basis for $\mathcal{O}$? e.i. something like \begin{equation} \Delta = \gcd \left\{ D\left( \omega_j \right) \right\}_{j=1}^{n-1}, \end{equation} which would claim that the indices $a_j$ of $\mathbb{Z}[\omega_j]$ (for $j \geq 1$) in $\mathcal{O}$ are relatively prime? If the equation set out above is false in general, what is known about $\gcd \left\{ a_j \right\}_{j=1}^{n-1}$?

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