Lagmental Vicfred

A Normal Basis Is One Frobenius Orbit by Vicfred

Last updated: Wed 10 October 2018

tags: mathematics, finite fields, Frobenius, normal bases

Every finite Galois extension of finite fields has a basis consisting of conjugates of a single element. I want the notation, the mechanism, and the failure mode visible at the same time.

The data

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.

$$ \mathcal B_\alpha=(\alpha,\alpha^q,\ldots,\alpha^{q^{n-1}}) $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \exists\alpha\in\mathbf F_{q^n}\ \text{such that }\mathcal B_\alpha\text{ is an }\mathbf F_q\text{-basis} $$

Derivation

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

$$ \det\begin{pmatrix}\alpha&\alpha^q&\cdots&\alpha^{q^{n-1}}\\\alpha^q&\alpha^{q^2}&\cdots&\alpha\\\vdots&\vdots&\ddots&\vdots\end{pmatrix}\ne0 $$

Invariant content

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathcal B_\alpha=(\alpha,\alpha^q,\ldots,\alpha^{q^{n-1}}),\\[5pt] \mathsf{C}\;&:\quad \exists\alpha\in\mathbf F_{q^n}\ \text{such that }\mathcal B_\alpha\text{ is an }\mathbf F_q\text{-basis}. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists\alpha\in\mathbf F_{q^n}\ \text{such that }\mathcal B_\alpha\text{ is an }\mathbf F_q\text{-basis} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 13 October 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.