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# Application of the Strong Artin Conjecture to the Class Number Problem

Published:2012-09-08
Printed: Dec 2013
• Peter J. Cho,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
• Henry H. Kim,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
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## Abstract

We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group $A_4, S_4$ and $S_5$. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying zero density result of Kowalski-Michel, we choose subfamilies of $L$-functions which are zero free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s=1$, and by the class number formula, we obtain the extreme class numbers.
 Keywords: class number, strong Artin conjecture
 MSC Classifications: 11R29 - Class numbers, class groups, discriminants 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}