bibkey: conrad2025cyclotomicextensions authors: Keith Conrad year: 2025 title: Cyclotomic extensions doi: null url: https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf claim: Complex roots of unity and the prime cyclotomic polynomial give the classical circular finite sum ingredients. strata_touched: [] license: citation-only triage: anchor

Cyclotomic extensions

The author text, §1, printed p.1, identifies the complex $m$th roots of unity as $e^{2\pi ij/m}$ for $0\le j<m$ and describes their equally spaced circle positions. Section 5 defines cyclotomic polynomials and gives ${\Phi }_p(T)=1+T+\cdots +T^{p-1}$ for prime $p$.

The associated elementary finite sum identity is $(1-z)(1+z+\cdots +z^{m-1})=1-z^m$. Thus any nontrivial $m$th root gives a zero sum, including $m=2,3,5$. The identity is polynomial and needs no division until it is evaluated at $z\ne 1$.

This note supplies the classical circular readout, not an identification between a FIB mode set and a polygon rotation action. A different grading of the same configurations has a different generating polynomial.