defined by $x\mapsto x\beta$ is nonsingular, so $[\mathfrak{A}_{L/K}   \beta : \mathcal{O}_K[G]   \beta]=[\mathfrak{A}_{L/K}: \mathcal{O}_K[G]]$. The result follows.

\[rmk:SeparateIndices\] Proposition \[productindex\] shows that every index $[\mathcal{O}_L : \mathcal{O}_K[G]   \alpha]$ differs from the corresponding index $[\mathcal{O}_L : \mathfrak{A}_{L/K}   \alpha]$ by a constant independent of $\alpha$. Combined with Proposition \[associatedfree\], this suggests that it may be more natural to study separately the two quantities $[\mathfrak{A}_{L/K} : \mathcal{O}_K[G]]$ and $\min_{\alpha \in \mathcal{O}_L} [\mathcal{O}_L : \mathfrak{A}_{L/K}   \alpha]$. Therefore our results can be also read as bounds on the minimum of $[\mathcal{O}_L : \mathfrak{A}_{L/K}   \alpha]$.

Index of $\mathcal{O}_K[G]$ in a maximal order {#indexmaximal}
----------------------------------------------

In the spirit of Remark \[rmk:SeparateIndices\] we now give a simple upper bound on $[\mathfrak{A}_{L/K} : \mathcal{O}_K[G]]$ by estimating the index of $\mathcal{O}_K[G]$ in a *maximal* $\mathcal{O}_K$-order $\mathfrak{M}$ in which it is contained.

We consider on $K[G]$ the bilinear form $(x,y) \mapsto \operatorname{tr}(xy)$, where we denote by $\operatorname{tr}(z)$ the trace of multiplication by $z$ on the $K$-vector space $K[G]$; it is nondegenerate since $K$ has characteristic $0$, so it allows us to compute discriminants of lattices in the sense of Definition \[def:Discriminant\]. Let $\underline{m} = [\mathfrak{M} : \mathcal{O}_K[G]]_{\mathcal{O}_K}$. By Proposition \[prop:disc\] we have an equality of ideals $$\label{eq:DiscIndex}
\underline{m}^2 \cdot \operatorname{disc} \mathfrak{M} = 