Representation of cyclotomic fields and their subfields

Abstract

Let $$\mathbb{K}$$ be a finite extension of a characteristic zero field $$\mathbb{F}$$ . We say that a pair of n × n matrices (A,B) over $$\mathbb{F}$$ represents $$\mathbb{K}$$ if $$\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle B \right\rangle }}$$ , where $$\mathbb{F}\left[ A \right]$$ denotes the subalgebra of $$\mathbb{M}_n \left( \mathbb{F} \right)$$ containing A and 〈B〉 is an ideal in $$\mathbb{F}\left[ A \right]$$ , generated by B. In particular, A is said to represent the field $$\mathbb{K}$$ if there exists an irreducible polynomial $$q\left( x \right) \in \mathbb{F}\left[ x \right]$$ which divides the minimal polynomial of A and $$\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {q\left( A \right)} \right\rangle }}$$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $$\mathbb{K}$$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $$\mathbb{K}$$ , 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 order 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.