Integral bases and Discriminants

[Firepanda]

Here is my solution, I think I have part i) done OK, but I'm not sure about how to proceed with part ii).

I suppose I need to show that both determinants of the base change matrices Cij and Dij are = ±1?Thanks

 

[morphism]

You pretty much have it... Just combine both your equations into
$$\Delta(\alpha_1,\ldots,\alpha_n) = (\det D_{ij})^2 (\det C_{ij})^2 \Delta(\alpha_1,\ldots,\alpha_n).$$
What can you conclude from this?

 

[Firepanda]

You pretty much have it... Just combine both your equations into
$$\Delta(\alpha_1,\ldots,\alpha_n) = (\det D_{ij})^2 (\det C_{ij})^2 \Delta(\alpha_1,\ldots,\alpha_n).$$
What can you conclude from this?

that $(\det D_{ij})^2 (\det C_{ij})^2 = 1$?

so the determinants are either ±1, so we can conclude the statement?

 

[morphism]

Yes, because one discriminant is (det)^2 times the other!