Discriminant (of a field basis)

Let $L/K$ be a finite field extension of degree $n$, and let $\mathrm{Tr}_{L/K}$ be the field trace . For an $n$-tuple $\mathbf{b}=(b_1,\dots,b_n)$ in $L$ that is a $K$-basis of $L$, the discriminant of $\mathbf{b}$ (relative to $L/K$) is

If $L/K$ is separable (see separable extension ), then $\mathrm{disc}_{L/K}(\mathbf{b})\neq 0$, and one can also express it using $K$-embeddings $\sigma_1,\dots,\sigma_n:L\hookrightarrow \Omega$ into a common overfield $\Omega$:

This shows how the discriminant measures “linear independence of conjugates” and links naturally to separability (compare separable elements have distinct conjugates ).

Examples

1. Quadratic basis. Let $L=K(\sqrt{d})$ with $\mathrm{char}(K)\neq 2$ and basis $(1,\sqrt{d})$. Using $\mathrm{Tr}_{L/K}(1)=2$, $\mathrm{Tr}_{L/K}(\sqrt{d})=0$, $\mathrm{Tr}_{L/K}(d)=2d$,

   $$\mathrm{disc}_{L/K}(1,\sqrt{d})=\det\begin{pmatrix}2&0\\0&2d\end{pmatrix}=4d.$$

2. Power basis in a simple extension. If $L=K(\alpha)$ with $[L:K]=n$, the “power basis” $(1,\alpha,\dots,\alpha^{n-1})$ has discriminant $\mathrm{disc}_{L/K}(1,\alpha,\dots,\alpha^{n-1})=\det(\mathrm{Tr}_{L/K}(\alpha^{i+j}))_{0\le i,j\le n-1}$, which can be computed from the minimal polynomial of $\alpha$ in concrete cases.

3. Finite fields: always zero over the prime field when inseparable is absent? For $L=\mathbb{F}_{q^n}$ over $K=\mathbb{F}_q$, the extension is separable because finite fields are perfect , so discriminants of $K$-bases are nonzero. For example, if $n=2$ and $L=K(\alpha)$ with $\alpha^2-\alpha-\beta=0$ irreducible over $K$, then for the basis $(1,\alpha)$,

   $$\mathrm{disc}_{L/K}(1,\alpha)=\det\begin{pmatrix} \mathrm{Tr}(1) & \mathrm{Tr}(\alpha)\\ \mathrm{Tr}(\alpha) & \mathrm{Tr}(\alpha^2) \end{pmatrix}\in K^\times.$$