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Representation of cyclotomic fields and their subfields
Authors:A Satyanarayana Reddy  Shashank K Mehta  Arbind K Lal
Institution:111. Department of Mathematics, Shiv Nadar University, Dadri, 203 207, India
211. Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, 208 016, India
311. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208 016, India
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 $\mathbb{K} \cong {{\mathbb{F}\left A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , 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 $\mathbb{K} \cong {{\mathbb{F}\left A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . 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.
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