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Regulators and class numbers of an infinite family of quintic function fields. (English) Zbl 1442.11156
The objective of this paper is the study of a certain infinite family $$\{K_h\}_{h\in A}$$ of quintic function fields assuming that the characteristic $$p$$ is different from $$5$$, where $$A={\mathbb F}_q[x]$$ and $$k={\mathbb F}_q(x)$$. In fact, the elements of the family $$\{K_h\}$$ are subfields of cyclotomic function fields, that have the same conductors. The authors find the system of fundamental units and regulators of the elements of $$\{K_h\}$$ (Theorem 1.1), obtaining a result on the divisibility of the class numbers of cyclotomic function fields (Theorem 1.2). In fact they find the ideal class number $$h({\mathcal O}_h)$$ of $$K_h$$ (Theorem 1.3).
One of the main tools is the use of the notion of Lagrange resolvents of the generating quintic polynomials $$F_h(x)$$ of $$K_h$$. From the Lagrange resolvents, it is determined the rank of the unit group of $$K_h$$. This unit rank is $$4$$ and in fact, $$K_h$$ is a totally real function field. The regulator and the system of fundamental units of $$K_h$$ are explicitly found. In fact the regulator $$R_h$$ of $$K_h$$ equals $$R_h=71 (\deg h)^4$$.
In the last section, it is shown that there are infinitely many irregular primes of second class $$f\in A$$ such that $$h(k(\Lambda_f)^+)\equiv 0\bmod p^4$$ where $$k(\Lambda_N)^+$$ denotes the real subfield of the cyclotomic function field $$k(\Lambda_N)$$, $$N\in A$$.
##### MSC:
 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) 11R29 Class numbers, class groups, discriminants 11R58 Arithmetic theory of algebraic function fields
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