Lagmental Vicfred

Multiplicative Hilbert 90 Describes Norm-One Elements by Vicfred

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

In a cyclic extension, an element has norm one exactly when it is a quotient sigma(y) over y. I will separate the object being defined from the consequence being claimed.

Definitions first

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.

$$ N_{L/K}(x)=1 $$

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.

$$ x=\frac{\sigma(y)}{y}\quad\text{for some }y\in L^\times $$

A small case in full

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ N\!\left(\frac{\sigma(y)}y\right)=\prod_{i=0}^{n-1}\frac{\sigma^{i+1}(y)}{\sigma^i(y)}=\frac{\sigma^n(y)}y=1 $$

The reusable statement

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad N_{L/K}(x)=1,\\[5pt] \mathsf{C}\;&:\quad x=\frac{\sigma(y)}{y}\quad\text{for some }y\in L^\times. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] x=\frac{\sigma(y)}{y}\quad\text{for some }y\in L^\times \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Fri 31 July 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.