Using discriminants to find order of extension Any hints on how to show $[G:H]^{2}=\frac{disc(H)}{disc(G)}$,where G,H are free abelian groups of rank n and $H\subset G\subset K$,where K is a number field?
Alternative formulation, how to relate $[R:Z[a]]$ and disc(R),disc(Z[a])?
thanks

I assume that what you mean is something to this effect:
If $\mathcal{O}\subseteq\mathcal{O}'$ are orders of $K$ (for some number field $K$) then $[\mathcal{O}':\mathcal{O}]\text{disc}(\mathcal{O}')=\text{disc}(\mathcal{O})$. This follows immediately from the following theorem of algebra:

Theorem: Let $R$ be a PID, and $M$ a free-module of rank $n$. Suppose that $N\leqslant M$ is a free module of rank $n$. Then, there exists a basis $\{b_1,\ldots,b_n\}$ of $M$ and $r_1,\ldots,r_n\in R$ such that $\{r_1b_1,\ldots,r_nb_n\}$ is a basis for $N$.

Do you see why? (Hint: what are the $r_i$?)
