an:06970424
Zbl 1442.11156
Lee, Jungyun; Lee, Yoonjin
Regulators and class numbers of an infinite family of quintic function fields
EN
Acta Arith. 185, No. 2, 107-125 (2018).
00419405
2018
j
11R60 11R29 11R58
regulator; function field; quintic extension; class number; cyclotomic function field; irregular prime; totally real cyclotomic function fields; ideal class numbers; Lagrange resolvents
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 \(k(\Lambda_N)\), \(N\in A\).
Gabriel D. Villa Salvador (Ciudad de M??xico)