Monogenic dihedral quartic extensions
type: Item
instance of: scholarly article
title: Monogenic dihedral quartic extensions
zbMATH Open document ID: 1462.11104
author: Henry H. Kim
published in: The Ramanujan Journal
publication date: 2019-12-06T00:00:00Z
review text: A number field whose ring of integers has a power integral basis of the form \(O_K= \mathbb{Z}[\vartheta]\), is called a monogenic number field. It is expected that monogenic number fields of a high degree, are very seldom. Define \(L^{(m)}(X)=\{\text{set}\) of monogenic fields of absolute discriminants \(\le X\) with Galois group \(S_n\) for its Galois closures\}. Much has been investigated regarding \(|L^{(m)}(X)|\). Bounds have been established for some explicit situations. As to monogenic \(D_4\)-quartic fields \(X\) (\(D_4\cong\) dihedral group of order 8), the main result of the paper is: Theorem 1.1 \(|L^{(m)}(X)|= O(X^{\frac{3}{4}}\cdot (\log X)^3)\). There are in this paper some other, be it somehow technical, results for such \(L^{(i)}(X)\) for some particular kinds of \(X\). It is interesting to learn some techniques in doing calculations in this paper.
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 7140973
zbMATH Keywords: monogenic extensions
zbMATH Keywords: dihedral quartic extensions
zbMATH Keywords: counting number field
zbMATH Keywords: discriminants of fields
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: s11139-018-0049-0
OpenAlex ID: W2890530953
cites work: Squarefree values of polynomial discriminants. I
cites work: Enumerating quartic dihedral extensions of \(\mathbb Q\) with signatures.
cites work: Q4550589
cites work: On the representation of a number as the sum of a square and a product
cites work: On the number of divisors of quadratic polynomials
cites work: Integral bases for quartic fields with quadratic subfields
cites work: Power bases in dihedral quartic fields
cites work: Asymptotics of number fields and the Cohen-Lenstra heuristics
cites work: Power free values of polynomials
cites work: On the number of integral ideals of given norm and ray‐class
cites work: Handbook of Number Theory I
Wikidata QID: Q129268662
DOI: 10.1007/S11139-018-0049-0
description: scientific article; zbMATH DE number 7140973