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Representation of cyclotomic fields and their subfields
, Mehta S.K., Lal A.K.
Published in
Volume: 44
Issue: 2
Pages: 203 - 230
Let K be a finite extension of a characteristic zero field F. We say that a pair of n × n matrices (A,B) over F represents K if K ≅ F[A]/〈B〉, where F[A] denotes the subalgebra of Mn(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) ∈ F[x] which divides the minimal polynomial of A and K ≅ F[A]/〈q(A)〉. 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 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 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. © 2013 The Indian National Science Academy.
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