Lagmental Vicfred

The Trace Pairing over a Finite Field Is Nondegenerate by Vicfred

Last updated: Tue 10 August 2021

tags: mathematics, finite fields, Frobenius, normal bases

The bilinear form Tr(xy) identifies a finite separable extension with its dual vector space. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

For \(q=p^r\), the Frobenius map \(F(x)=x^q\) controls extensions of \(\mathbf F_q\). Its orbits determine minimal polynomials, trace, norm, and the Galois group.

$$ \langle x,y\rangle=\operatorname{Tr}_{\mathbf F_{q^n}/\mathbf F_q}(xy) $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ x\ne0\Longrightarrow\exists y,\ \operatorname{Tr}(xy)\ne0 $$

A small case in full

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ G_{ij}=\operatorname{Tr}(\alpha_i\alpha_j),\qquad\det(G_{ij})\ne0 $$

The reusable statement

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \langle x,y\rangle=\operatorname{Tr}_{\mathbf F_{q^n}/\mathbf F_q}(xy),\\[5pt] \mathsf{C}\;&:\quad x\ne0\Longrightarrow\exists y,\ \operatorname{Tr}(xy)\ne0. \end{aligned} $$

A nearby false statement

Frobenius is \(\mathbf F_q\)-linear on an extension but not generally linear over a larger coefficient field. Exponents must match the chosen base.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] x\ne0\Longrightarrow\exists y,\ \operatorname{Tr}(xy)\ne0 \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Thu 21 December 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.