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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.

##### MSC:
 11C08 Polynomials in number theory 11C20 Matrices, determinants in number theory 11R18 Cyclotomic extensions
##### Keywords:
circulant matrix; polynomial cycle; cyclotomic ield
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##### References:
 [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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