Lagmental Vicfred

The Derivative Detects Repeated Irreducible Factors by Vicfred

tags: mathematics, splitting fields, separability, Galois theory

A polynomial is separable when it is coprime to its formal derivative. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Objects and notation

A finite extension \(L/K\) is Galois when it is both normal and separable. Its group \(G=\operatorname{Gal}(L/K)\) records all automorphisms fixing \(K\).

$$ f'(x)=\sum_{i\ge1}ia_ix^{i-1} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ f\ \text{separable}\Longleftrightarrow\gcd(f,f')=1 $$

Push the symbols

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

$$ \operatorname{char}K=p,\qquad f'(x)=0\Longleftrightarrow f(x)=\sum_j a_jx^{pj} $$

Structural reading

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f'(x)=\sum_{i\ge1}ia_ix^{i-1},\\[5pt] \mathsf{C}\;&:\quad f\ \text{separable}\Longleftrightarrow\gcd(f,f')=1. \end{aligned} $$

A hypothesis worth keeping

Normal and separable are independent hypotheses outside perfect fields. Having the right degree alone does not make an extension Galois.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f\ \text{separable}\Longleftrightarrow\gcd(f,f')=1 \end{gathered}} $$

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 Wed 20 September 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.