Representation of Cyclotomic Fields and Their Subfields

Let $\text{K}$ be a finite extension of a characteristic zero field $\text{F}$. We say that the pair of $n\times n$ matrices $(A,B)$ over $\text{F}$ represents $\text{K}$ if $\K \cong \F[A]/< B >$where$\F[A]$denotesthesmallestsubalgebraof$M_n(\F)$containing$A$and$< B >$isanidealin$\F[A]$generatedby$B$. In particular, $A$ is said to represent the field $\text{K}$ if there exists an irreducible polynomial $q(x)\in \text{F}[x]$ which divides the minimal polynomial of $A$ and $\text{K}\cong \text{F}[A]\mathrm{/}<q(A)>$. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and $\text{K}$ 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 $(A,\text{J})$ represents $\text{K}$, where $\text{J}$ 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.