Lagmental Vicfred

An Irreducible Cubic Has Galois Group A3 or S3 by Vicfred

tags: mathematics, polynomials, discriminants, Galois groups

For an irreducible separable cubic, the discriminant decides between the two transitive possibilities. I will separate the object being defined from the consequence being claimed.

Statement

For a separable polynomial \(f\in K[x]\), the Galois group permutes its roots faithfully. Factorisations, discriminants, and resolvents constrain the resulting subgroup of \(S_n\).

$$ f(x)\in K[x]\ \text{irreducible},\qquad\deg f=3 $$

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.

$$ \operatorname{Gal}(f/K)\cong\begin{cases}A_3,&\Delta(f)\in(K^\times)^2,\\S_3,&\Delta(f)\notin(K^\times)^2,\end{cases} $$

Worked algebra

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

$$ f(x)=x^3-2,\qquad\Delta(f)=-108=-3\cdot6^2,\qquad\operatorname{Gal}(f/\mathbf Q)\cong S_3 $$

Conceptual compression

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(x)\in K[x]\ \text{irreducible},\qquad\deg f=3,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Gal}(f/K)\cong\begin{cases}A_3,&\Delta(f)\in(K^\times)^2,\\S_3,&\Delta(f)\notin(K^\times)^2,\end{cases}. \end{aligned} $$

Caveat

A discriminant square distinguishes containment in \(A_n\), but it usually does not determine the entire Galois group by itself.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Gal}(f/K)\cong\begin{cases}A_3,&\Delta(f)\in(K^\times)^2,\\S_3,&\Delta(f)\notin(K^\times)^2,\end{cases} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sun 26 June 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.