Lagmental Vicfred

Galois Correspondence Reverses Inclusion by Vicfred

tags: mathematics, splitting fields, separability, Galois theory

Larger subgroups fix smaller intermediate fields and normal subgroups correspond to Galois intermediate extensions. The point is to make the formal expression readable enough to audit line by line.

Statement

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 \(K\).

$$ H\longmapsto L^H,\qquad F\longmapsto\operatorname{Gal}(L/F) $$

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.

$$ H_1\le H_2\Longrightarrow L^{H_2}\subseteq L^{H_1} $$

Worked algebra

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \begin{array}{c|c|c}H\trianglelefteq G&L^H/K\text{ Galois}&\operatorname{Gal}(L^H/K)\cong G/H\end{array} $$

Conceptual compression

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad H\longmapsto L^H,\qquad F\longmapsto\operatorname{Gal}(L/F),\\[5pt] \mathsf{C}\;&:\quad H_1\le H_2\Longrightarrow L^{H_2}\subseteq L^{H_1}. \end{aligned} $$

Caveat

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] H_1\le H_2\Longrightarrow L^{H_2}\subseteq L^{H_1} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sun 24 November 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.