31,310 research outputs found
th power residue chains of global fields
In 1974, Vegh proved that if is a prime and a positive integer, there
is an term permutation chain of th power residue for infinitely many
primes [E.Vegh, th power residue chains, J.Number Theory, 9(1977), 179-181].
In fact, his proof showed that is an term permutation
chain of th power residue for infinitely many primes. In this paper, we
prove that for any "possible" term sequence , there are
infinitely many primes making it an term permutation chain of th
power residue modulo , where is an arbitrary positive integer [See
Theorem 1.2]. From our result, we see that Vegh's theorem holds for any
positive integer , not only for prime numbers. In fact, we prove our result
in more generality where the integer ring is replaced by any -integer
ring of global fields (i.e. algebraic number fields or algebraic function
fields over finite fields).Comment: 4 page
The genus fields of Artin-Schreier extensions
Let be a power of a prime number . Let be the
rational function field with constant field . Let
be an Artin-Schreier extension of . In this paper, we explicitly describe
the ambiguous ideal classes and the genus field of . Using these results we
study the -part of the ideal class group of the integral closure of
in . And we also give an analogy of
Rdei-Reichardt's formulae for .Comment: 9 pages, Corrected typo
Monomial Dynamical Systems of Dimension One over Finite Fields
In this paper we study the monomial dynamical systems of dimension one over
finite fields from the viewpoints of arithmetic and graph theory. We give
formulas for the number of periodic points with period r and cycles with length
r. Then we compute the natural distributions of periodic points and cycles. We
also define and compute the Dirichlet distributions of periodic points and
cycles. Especially, we associate the monomial dynamical systems with function
fields to compute distributions.Comment: Accepted by Acta Arithmetica. (SCI
On hypergeometric Bernoulli numbers and polynomials
In this note, we shall provide several properties of hypergeometric Bernoulli
numbers and polynomials, including sums of products identity, differential
equations and recurrence formulas.Comment: 12 page
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