Lagmental Vicfred

Additive Hilbert 90 Describes Trace-Zero Elements by Vicfred

tags: mathematics, Kummer theory, Artin--Schreier theory, Hilbert 90

In a cyclic extension, trace-zero elements are differences sigma(y) minus y. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The data

Kummer equations \(x^n=a\) describe cyclic extensions when roots of unity are available and \(\operatorname{char}K\nmid n\). In characteristic \(p\), Artin--Schreier equations \(x^p-x=a\) play the parallel role.

$$ L/K\ \text{cyclic},\qquad G=\langle\sigma\rangle $$

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.

$$ \operatorname{Tr}_{L/K}(x)=0\Longleftrightarrow x=\sigma(y)-y\ \text{for some }y\in L $$

Derivation

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \operatorname{Tr}(\sigma(y)-y)=\sum_{i=0}^{n-1}\bigl(\sigma^{i+1}(y)-\sigma^i(y)\bigr)=0 $$

Invariant content

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad L/K\ \text{cyclic},\qquad G=\langle\sigma\rangle,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Tr}_{L/K}(x)=0\Longleftrightarrow x=\sigma(y)-y\ \text{for some }y\in L. \end{aligned} $$

Scope

Both theories have hypotheses that cannot be removed casually. Missing roots of unity or inseparability changes the Galois group and the classification.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Tr}_{L/K}(x)=0\Longleftrightarrow x=\sigma(y)-y\ \text{for some }y\in L \end{gathered}} $$

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 24 October 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.