18,890 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
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