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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 \(k(\Lambda_N)\), \(N\in A\).
11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
11R29 Class numbers, class groups, discriminants
11R58 Arithmetic theory of algebraic function fields
Full Text: DOI
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