Representation of Cyclotomic Fields and Their Subfields

TL;DR

This paper investigates minimal matrix representations of subfields within cyclotomic fields, providing explicit constructions using circulant and companion matrices, and establishing bounds based on the prime factorization of the field degree.

Contribution

It identifies the smallest circulant-matrix representations for subfields of cyclotomic fields and constructs explicit zero-one circulant matrices for prime cyclotomic fields, also providing bounds for more complex cases.

Findings

Constructed minimal circulant matrices for subfields of cyclotomic fields.

Provided explicit zero-one circulant matrices for prime cyclotomic fields.

Established bounds on the size of companion matrices for cyclotomic fields with multiple prime factors.

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.