Lagmental Vicfred

Normal Extensions Contain Every Conjugate by Vicfred

tags: mathematics, splitting fields, separability, Galois theory

A finite extension is normal exactly when every irreducible polynomial with one root inside splits completely inside. This is a compact note, but the quantifiers and hypotheses stay on the page.

Set-up

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 \(K\).

$$ L/K\ \text{normal} $$

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.

$$ m_{\alpha,K}\ \text{has one root in }L\Longrightarrow m_{\alpha,K}\ \text{splits in }L $$

The calculation

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \mathbf Q(\sqrt[3]2)/\mathbf Q\ \text{is not normal},\qquad \zeta_3\sqrt[3]2\notin\mathbf R\supset\mathbf Q(\sqrt[3]2) $$

What survives abstraction

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad L/K\ \text{normal},\\[5pt] \mathsf{C}\;&:\quad m_{\alpha,K}\ \text{has one root in }L\Longrightarrow m_{\alpha,K}\ \text{splits in }L. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] m_{\alpha,K}\ \text{has one root in }L\Longrightarrow m_{\alpha,K}\ \text{splits in }L \end{gathered}} $$

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