Discriminant of a field extension

Summary

The discriminant of a basis $b_1, \ldots, b_n$ of a finite extension $L/K$ is the determinant of the matrix $\left(\mathrm{Tr}_{L/K}(b_i b_j)\right)$, the Gram matrix of the trace form. Changing the basis multiplies it by the square of a nonzero element of $K$, so its class modulo squares depends only on the extension. It is nonzero exactly when $L/K$ is separable. For a power basis $1, \alpha, \ldots, \alpha^{n-1}$ it equals the discriminant of the minimal polynomial of $\alpha$. For an integral basis of a number field it gives the discriminant of the number field.