Lagmental Vicfred

The Fixed Field of the Full Galois Group Is the Base by Vicfred

tags: mathematics, splitting fields, separability, Galois theory

For a finite Galois extension, elements fixed by every automorphism are exactly the base field. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

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 \(K\).

$$ G=\operatorname{Gal}(L/K),\qquad L^G=\{x\in L:\sigma x=x\ \forall\sigma\in G\} $$

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 \(\Longleftrightarrow\).

$$ L^G=K $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ [L:L^H]=|H|,\qquad H=\operatorname{Gal}(L/L^H)\quad(H\le G) $$

The reusable statement

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

$$ \begin{aligned} \mathsf{D}\;&:\quad G=\operatorname{Gal}(L/K),\qquad L^G=\{x\in L:\sigma x=x\ \forall\sigma\in G\},\\[5pt] \mathsf{C}\;&:\quad L^G=K. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] L^G=K \end{gathered}} $$

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 Wed 17 June 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.