Lagmental Vicfred

Trace and Norm Are Additive and Multiplicative Shadows by Vicfred

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

Trace linearises addition while norm multiplies determinants of multiplication operators. I will separate the object being defined from the consequence being claimed.

Start locally

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.

$$ m_\alpha:L\to L,\qquad x\mapsto\alpha x $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \operatorname{Tr}_{L/K}(\alpha)=\operatorname{tr}(m_\alpha),\qquad N_{L/K}(\alpha)=\det(m_\alpha) $$

Compute before generalising

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

$$ \begin{pmatrix}a&bd\\b&a\end{pmatrix}\quad\Longrightarrow\quad\begin{cases}\operatorname{Tr}(a+b\sqrt d)=2a,\\N(a+b\sqrt d)=a^2-db^2.\end{cases} $$

The global view

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad m_\alpha:L\to L,\qquad x\mapsto\alpha x,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Tr}_{L/K}(\alpha)=\operatorname{tr}(m_\alpha),\qquad N_{L/K}(\alpha)=\det(m_\alpha). \end{aligned} $$

Edge conditions

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Tr}_{L/K}(\alpha)=\operatorname{tr}(m_\alpha),\qquad N_{L/K}(\alpha)=\det(m_\alpha) \end{gathered}} $$

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 Thu 17 October 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.