An analogue of Girstmair's formula in function fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-01-27 DOI:10.1016/j.ffa.2025.102585

Daisuke Shiomi

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引用次数: 0

Abstract

Suppose that p is an odd prime and

g > 1

is a primitive root modulo p. Let M be a number field contained in the p-th cyclotomic field. In 1994, Girstmair found a surprising relation between the relative class number of M and the digits of

1 / p

in base g. In this paper, we consider an analogue of Girstmair's formula in function fields. Suppose that

P \in F_{q} [T]

is monic irreducible and

G \in F_{q} [T]

is a primitive root modulo P. Let L be a field extension of

F_{q} (T)

which is contained in the P-th cyclotomic function field. Our goal is to give relations between the plus and minus parts of the divisor class number of L and the digits of

1 / P

in base G.

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.