# Dihedral and cyclic extensions with large class numbers

@article{Cho2012DihedralAC, title={Dihedral and cyclic extensions with large class numbers}, author={Peter J. Cho and Henry H. Kim}, journal={Journal de Theorie des Nombres de Bordeaux}, year={2012}, volume={24}, pages={583-603} }

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…

## 8 Citations

Application of the Strong Artin Conjecture to the Class Number Problem

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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…

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It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get…

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## References

SHOWING 1-10 OF 29 REFERENCES

Non-abelian number fields with very large class numbers

- Mathematics
- 2006

1.1. Background and motivation. Let K be a number field and denote by H its group of ideal classes. Since H is finite an interesting question one may ask is how its size, the class number of K,…

Class numbers of the simplest cubic fields

- Mathematics
- 1987

Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n.…

Unit groups and class numbers of real cyclic octic fields

- Mathematics
- 1991

The generating polynomials of D. Shanks' simplest quadratic and cubic fields and M.-N. Grass simplest quartic and sextic fields can be obtained by working in the group PGL2(Q) . Following this…

Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

- Mathematics
- 2002

This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is…

Zeros of families of automorphic $L$-functions close to 1

- Mathematics
- 2002

For many L-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number…

Special units in real cyclic sextic fields

- Mathematics
- 1987

We study the real cyclic sextic fields generated by a root w of (X — l)6 (r2 + 108)(X2 + A')2, t e Z {0, ±6, ±26}. We show that, when t1 + 108 is square-free (except for powers of 2 and 3), and t =t…

DIHEDRAL QUINTIC FIELDS WITH A POWER BASIS

- Mathematics
- 2005

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,…

Certain Quartic Fields with Small Regulators

- Mathematics
- 1996

Abstract For quartic fields with a quadratic subfield, explicit lower bounds of the regulators are given in terms of the discriminant. Further, these bounds are shown to be in a sense best possible…