A note on polynomial cycles. (English) Zbl 1053.11021

Let \(p\) be a prime, and let \(C_p\) be the ring of circulant matrices with rational integral entries and order \(p\). The author establishes a connection between cycles of mappings induced by a polynomial \(f\in{\mathbb Z}[X]\) in \(C_p\), and the cycles induced by \(f\) in the ring of integers of the \(p\)-th cyclotomic field.


11C08 Polynomials in number theory
11C20 Matrices, determinants in number theory
11R18 Cyclotomic extensions
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[1] Boduch J.: Polynomial cycles in rings of algebraic integer. (Polish) MA thesis Wroclaw University, 1990.
[2] Davis P. J.: Circulant Matrices. Wiley-Interscience publishers, John Wiley and sons, New York-Chichester -Brisbane -Toronto, 1979. · Zbl 0418.15017
[3] Divišová Z.: On cycles of polynomials with integral rational coefficients. Mathematica Slovaca) · Zbl 1028.11064
[4] Halter-Koch F., Konečná P.: Polynomial cycles in finite extension fields. Mathematica Slovaca) · Zbl 1028.11014
[5] Kostra J.: A note on representation of cyclotomic fields. Acta Mathematica et Informatica Universitatis Ostraviensis 4, 29-35, 1996. · Zbl 0870.11068
[6] Kostra J.: On orbits in ambiguons ideals. Acta Acad. Paed. Agriensis, Section Mathematical) · Zbl 1017.11012
[7] Narkiewicz W.: Polynomial Mappings. Lecture Notes in Mathematics, 1600, Springer-Verlag, Berlin, Heidelberg, 1995. · Zbl 0829.11002
[8] Pomp M., Havelek R.: On representation of cyclotomic fields \(\mathbb Q (\zeta_{pq}). Acta Mathematica et Informatica Universitatis Ostraviensis, 7, 71-78, 1999.\) · Zbl 1024.11067
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