Lagmental Vicfred

The Discriminant Changes by the Sign of a Root Permutation by Vicfred

tags: mathematics, polynomials, discriminants, Galois groups

The Vandermonde product squares to the discriminant, so odd permutations negate its square root. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Objects and notation

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

$$ \Delta(f)=a_n^{2n-2}\prod_{i<j}(\alpha_i-\alpha_j)^2 $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \sigma\!\left(a_n^{n-1}\prod_{i<j}(\alpha_i-\alpha_j)\right)=\operatorname{sgn}(\sigma)\sqrt{\Delta(f)} $$

Push the symbols

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

$$ \det\begin{pmatrix}1&\alpha_1&\cdots&\alpha_1^{n-1}\\1&\alpha_2&\cdots&\alpha_2^{n-1}\\\vdots&\vdots&\ddots&\vdots\\1&\alpha_n&\cdots&\alpha_n^{n-1}\end{pmatrix}=\prod_{i<j}(\alpha_j-\alpha_i) $$

Structural reading

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad \Delta(f)=a_n^{2n-2}\prod_{i<j}(\alpha_i-\alpha_j)^2,\\[5pt] \mathsf{C}\;&:\quad \sigma\!\left(a_n^{n-1}\prod_{i<j}(\alpha_i-\alpha_j)\right)=\operatorname{sgn}(\sigma)\sqrt{\Delta(f)}. \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \sigma\!\left(a_n^{n-1}\prod_{i<j}(\alpha_i-\alpha_j)\right)=\operatorname{sgn}(\sigma)\sqrt{\Delta(f)} \end{gathered}} $$

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 26 March 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.