×

Monogenic dihedral quartic extensions. (English) Zbl 1462.11104

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.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11M41 Other Dirichlet series and zeta functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bhargava, M., Shankar, A., Wang, X.: Squarefree values of polynomial discriminants I. arXiv:1611.09806 · Zbl 1492.11150
[2] Cohen, H.: Enumerating quartic dihedral extensions of \[{\mathbb{Q}}\] Q with signatures. Ann. L’Institute Fourier 53, 339-377 (2003) · Zbl 1114.11085
[3] Cohen, H., Diaz Y Diaz, F., Olivier, M.: Enumerating quartic dihedral extensions of \[{\mathbb{Q}}\] Q. Compos. Math 133, 65-93 (2002) · Zbl 1050.11104
[4] Hooley, C.: On the representation of a number as the sum of a square and a product. Math. Z. 69, 211-227 (1958) · Zbl 0081.03904
[5] Hooley, C.: On the number of divisors of quadratic polynomials. Acta Math. 110, 97-114 (1963) · Zbl 0116.03802
[6] Huard, J.G., Spearman, B.K., Williams, K.S.: Integral bases for quartic fields with quadratic subfields. J. Number Theory 51, 87-102 (1995) · Zbl 0826.11048
[7] Kable, A.: Power bases in dihedral quartic fields. J. Number Theory 76, 120-129 (1999) · Zbl 0934.11051
[8] Klüners, J.: Asymptotics of number fields and the Cohen-Lenstra heuristics. J. Theor. Nombres Bordeaux 18, 607-615 (2006) · Zbl 1142.11078
[9] Nair, M.: Power free values of polynomials. Mathematika 23, 159-183 (1976) · Zbl 0349.10039
[10] Odoni, R.W.K.: On the number of integral ideals of given norm and ray-class. Mathematika 38(1), 185-190 (1991) · Zbl 0754.11035
[11] Sandor, J., Mitrinovic, D.S., Crstici, B.: Handbook of Number Theory I. Springer, Heidelberg (2006) · Zbl 1151.11300
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.