Lagmental Vicfred

Cyclotomic Polynomials Have Abelian Galois Groups by Vicfred

tags: mathematics, polynomials, discriminants, Galois groups

The splitting field of the nth cyclotomic polynomial has Galois group the units modulo n. I will separate the object being defined from the consequence being claimed.

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 \(S_n\).

$$ K=\mathbf Q(\zeta_n),\qquad\sigma_a(\zeta_n)=\zeta_n^a $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \operatorname{Gal}(K/\mathbf Q)\cong(\mathbf Z/n\mathbf Z)^\times $$

Stress the formula

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

$$ n=8:\qquad(\mathbf Z/8\mathbf Z)^\times=\{1,3,5,7\}\cong C_2\times C_2 $$

Interpretation

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad K=\mathbf Q(\zeta_n),\qquad\sigma_a(\zeta_n)=\zeta_n^a,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Gal}(K/\mathbf Q)\cong(\mathbf Z/n\mathbf Z)^\times. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Gal}(K/\mathbf Q)\cong(\mathbf Z/n\mathbf Z)^\times \end{gathered}} $$

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 Thu 14 September 2023. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.