Lagmental Vicfred

Artin--Schreier Extensions Are Additive by Vicfred

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

In characteristic p, a root of x^p minus x minus a has conjugates obtained by adding elements of F_p. The point is to make the formal expression readable enough to audit line by line.

Notation

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.

$$ f_a(x)=x^p-x-a,\qquad\operatorname{char}K=p $$

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.

$$ \alpha\ \text{root}\Longrightarrow\{\alpha+c:c\in\mathbf F_p\}\ \text{are all roots} $$

Stress the formula

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

$$ \sigma_c(\alpha)=\alpha+c,\qquad\sigma_c\sigma_d=\sigma_{c+d},\qquad\operatorname{Gal}(L/K)\cong(\mathbf F_p,+) $$

Interpretation

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_a(x)=x^p-x-a,\qquad\operatorname{char}K=p,\\[5pt] \mathsf{C}\;&:\quad \alpha\ \text{root}\Longrightarrow\{\alpha+c:c\in\mathbf F_p\}\ \text{are all roots}. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \alpha\ \text{root}\Longrightarrow\{\alpha+c:c\in\mathbf F_p\}\ \text{are all roots} \end{gathered}} $$

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 Sat 08 August 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.