On representation of cyclotomic fields \({\mathbb Q} (\zeta _{pq})\). (English) Zbl 1024.11067

The correspondence between circulant matrices of degree \(pq\) with \(p,q\) primes and elements of cyclotomic fields \({\mathbb Q}(\zeta _{pq})\) is found. This correspondence is via the determinant of a circulant matrix and the norm of an element of \({\mathbb Q}(\zeta _{pq})\). This correspondence gives a representation for elements of given subfields of the cyclotomic field \({\mathbb Q}(\zeta _{pq})\) by matrices.


11R18 Cyclotomic extensions
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[1] Borevich Z. I, Shafarevich I. R.: Number Theory. Moscow, 1964. · Zbl 0121.04202
[2] Davis P. J.: Circulant matrices. J. Wiley & Sons, Inc., New York, 1979. · Zbl 0418.15017
[3] Jakubec S., Kostra J., Nemoga K.: On the existence of an integral normal basis generated by a unit in prime extensions of rational numbers. Mathematics of computation 56 (1991), no. 194, 809-815. · Zbl 0722.11056
[4] Jakubec S., Kostra J.: A note on normal bases of ideals. Math. Slovaca 42 (1992), no. 5, 677-684. · Zbl 0773.11070
[5] Jakubec S., Kostra J.: On the Existence of a Normal Basis for an Ambiguous Ideal. Atti Sem. Mat. Fis. Univ. Modena XLVI (1998), 125-129. · Zbl 0912.11044
[6] Kostra J.: A Note on Representation of Cyclotomic Fields. Acta Math. Inf. Univ. Ostraviensis 4 (1996), 29-35. · Zbl 0870.11068
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