A SURVEY ON CYCLOTOMIC SUBFIELDS

Abstract

This paper is a survey on cyclotomic subfields and an improved version of the author's work in [1-5]. We show a relationship between cyclotomic and Dickson polynomials with polynomials of the form$$R_n(x)=\prod_{(i, n)=1}^{[(n / 2)]}\left(x-\varsigma_n^i-\varsigma_n^{-i}\right) .$$Based on these results, we show that $\mathbb{Q}\left(\varsigma_d+\varsigma_d^{-1}\right) \mid \Lambda=\mathbb{Z}\left[\varsigma_d+\varsigma_d^{-1}\right]$, where $\Lambda$ denotes the ring of algebraic integers. Given a divisor $d$ of $\left[\mathbb{Q}\left(\varsigma_m\right): \mathbb{Q}\right](m$ odd $)$, we also determine an algebraic integer $\alpha$ generating a subfield $F$ of degree $d$ over $\mathbb{Q}$, providing explicitly the minimum polynomial of $\alpha$ for the cases $d=2$ and $d=\phi(m) / 2$. Received: August 21, 2023Accepted: September 19, 2023