Discriminant of a field basis

Definition. Let $L/K$ be a finite separable extension of degree $n$, and let $\alpha_1,\dots,\alpha_n$ be a $K$-basis of $L$. The discriminant of this basis is

where $\mathrm{Tr}_{L/K}$ is the field trace and $\det$ is the determinant .

Under a change of basis by a matrix $M\in \mathrm{GL}_n(K)$, the discriminant scales by $(\det M)^2$. In particular, whether the discriminant is zero or nonzero does not depend on the chosen basis (for separable extensions it is nonzero).

See also. field norm , degree .

Examples.

1. For $L=\mathbb{Q}(\sqrt2)$ with basis $(1,\sqrt2)$: $\big(\mathrm{Tr}(\alpha_i\alpha_j)\big)= \begin{pmatrix} 2 & 0\\ 0 & 4 \end{pmatrix} \quad\Rightarrow\quad \mathrm{disc}(1,\sqrt2)=8.$
2. For $L=\mathbb{Q}(i)$ with basis $(1,i)$, one gets $\mathrm{disc}(1,i)=-4$.
3. For $L=\mathbb{F}_{q^2}$ over $\mathbb{F}_q$, any $\mathbb{F}_q$-basis has nonzero discriminant because the trace pairing $(x,y)\mapsto \mathrm{Tr}(xy)$ is nondegenerate.