Abstract
In this paper, we prove that the number of monogenic dihedral quartic extensions of absolute discriminants \(\le X\) is of size \(O(X^{\frac{3}{4}}(\log X)^3)\).
Similar content being viewed by others
References
Bhargava, M., Shankar, A., Wang, X.: Squarefree values of polynomial discriminants I. arXiv:1611.09806
Cohen, H.: Enumerating quartic dihedral extensions of \({\mathbb{Q}}\) with signatures. Ann. L’Institute Fourier 53, 339–377 (2003)
Cohen, H., Diaz Y Diaz, F., Olivier, M.: Enumerating quartic dihedral extensions of \( {\mathbb{Q}}\). Compos. Math 133, 65–93 (2002)
Hooley, C.: On the representation of a number as the sum of a square and a product. Math. Z. 69, 211–227 (1958)
Hooley, C.: On the number of divisors of quadratic polynomials. Acta Math. 110, 97–114 (1963)
Huard, J.G., Spearman, B.K., Williams, K.S.: Integral bases for quartic fields with quadratic subfields. J. Number Theory 51, 87–102 (1995)
Kable, A.: Power bases in dihedral quartic fields. J. Number Theory 76, 120–129 (1999)
Klüners, J.: Asymptotics of number fields and the Cohen–Lenstra heuristics. J. Theor. Nombres Bordeaux 18, 607–615 (2006)
Nair, M.: Power free values of polynomials. Mathematika 23, 159–183 (1976)
Odoni, R.W.K.: On the number of integral ideals of given norm and ray-class. Mathematika 38(1), 185–190 (1991)
Sandor, J., Mitrinovic, D.S., Crstici, B.: Handbook of Number Theory I. Springer, Heidelberg (2006)
Acknowledgements
We thank the referee for pointing out an error in the previous version.
Author information
Authors and Affiliations
Corresponding author
Additional information
This study was partially supported by an NSERC grant #482564.
Rights and permissions
About this article
Cite this article
Kim, H.H. Monogenic dihedral quartic extensions. Ramanujan J 50, 459–464 (2019). https://doi.org/10.1007/s11139-018-0049-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-018-0049-0