Representation of cyclotomic fields and their subfields

Let be a finite extension of a characteristic zero field . We say that a pair of n × n matrices ( A,B ) over represents if , where denotes the subalgebra of containing A and 〈 B 〉 is an ideal in , generated by B . In particular, A is said to represent the field if there exists an irreducible polynom...

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Bibliographic Details
Published inIndian journal of pure and applied mathematics Vol. 44; no. 2; pp. 203 - 230
Main Authors Reddy, A. Satyanarayana, Mehta, Shashank K., Lal, Arbind K.
Format Journal Article
LanguageEnglish
Published India Springer-Verlag 01.04.2013
Subjects
Applications of Mathematics
Mathematics
Mathematics and Statistics
Numerical Analysis
Companion Matrix
Cyclotomic field
Cyclotomic Polynomial
Möbius Function
Ramanujan Sum
Circulant matrix
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Summary:Let be a finite extension of a characteristic zero field . We say that a pair of n × n matrices ( A,B ) over represents if , where denotes the subalgebra of containing A and 〈 B 〉 is an ideal in , generated by B . In particular, A is said to represent the field if there exists an irreducible polynomial which divides the minimal polynomial of A and . 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 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 , 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.
ISSN:0019-5588
0975-7465
DOI:10.1007/s13226-013-0011-1