Representation of Cyclotomic Fields and Their Subfields

Abstract

Let $\K$ be a finite extension of a characteristic zero field $\F$. We say that the pair of $n\times n$ matrices $(A,B)$ over $\F$ represents $\K$ if $\K \cong \F[A]/< B >$ where $\F[A]$ denotes the smallest subalgebra of $M_n(\F)$ containing $A$ and $< B >$ is an ideal in $\F[A]$ generated by $B$. In particular, $A$ is said to represent the field $\K$ if there exists an irreducible polynomial $q(x)\in \F[x]$ which divides the minimal polynomial of $A$ and $\K \cong \F[A]/< q(A) >$. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and $\K$ is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that $(A,\J)$ represents $\K$, where $\J$ is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.