Representation of cyclotomic fields and their subfields

A. Satyanarayana Reddy (), Shashank K. Mehta () and Arbind K. Lal ()
Additional contact information
A. Satyanarayana Reddy: Shiv Nadar University
Shashank K. Mehta: Indian Institute of Technology
Arbind K. Lal: Indian Institute of Technology

Indian Journal of Pure and Applied Mathematics, 2013, vol. 44, issue 2, 203-230

Abstract: 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.

Keywords: Circulant matrix; Companion Matrix; Cyclotomic field; Cyclotomic Polynomial; Möbius Function; Ramanujan Sum (search for similar items in EconPapers)
Date: 2013
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-013-0011-1 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:44:y:2013:i:2:d:10.1007_s13226-013-0011-1

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-013-0011-1

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().