Relative Pólya group and Pólya dihedral extensions of Q

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$A_4$-Sextic Fields with a Power Basis

This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups Dn, n = 3, 4, 5, and cyclic groups Cn, n = 4, 5, 6. We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding L-functions 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 large class numbers.

/pdf/dihedral-and-cyclic-extensions-with-large-class-numbers-39uaq7whci.pdf

Dihedral and cyclic extensions with large class numbers

We give a sufficient criterion for specializations of certain families of polynomials to yield monogenic number fields. This generalizes constructions in several earlier papers. As applications we give new infinite families of monogenic number fields for several prescribed Galois groups.

A note on families of monogenic number fields

It is shown that there exist infinitely many PSLð2; 5Þ sextic fields with a power basis.

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PSL(2, 5) sextic fields with a power basis

to be monogenic. Dummit and Kisilevsky[4] have shown that there exist inﬁnitely many cyclic cubic ﬁelds whichare monogenic. The same has been shown for non-cyclic cubic ﬁelds, purequartic ﬁelds, bicyclic quartic ﬁelds, dihedral quartic ﬁelds by Spearman andWilliams [15], Funakura [6], Nakahara [14], Huard, Spearman and Williams[10] respectively. It is not known if there are inﬁnitely many monogeniccyclic quartic ﬁelds. If

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Melissa J. Lavallee

Papers

Dihedral quintic fields with a power basis