The proportion of cyclic quartic fields with discriminant divisible by a given prime. (English) Zbl 1068.11069

The authors determine an asymptotic formula for the number \(N_q(x)\) of cyclic quartic fields \(K\) with discriminant \(d(K)\leq x\) and \(d(K)\equiv 0\mod q\), where \(q\) is a given prime. They show that \[ N_q(x)=E_qx^{\frac12}+O(x^{\frac13}\log^3x), \] where \(E_q\) is an explicitly given constant. The paper is a continuation of an earlier one by Z. M. Ou and the second author [Can. Math. Bull. 44, No. 1, 97–104 (2001; Zbl 0996.11068)]. When writing these papers, the authors have obviously been unaware of the thesis of Sirpa Mäki [Ann. Acad. Sci. Fenn., Ser. A I, Diss. 54 (1985; Zbl 0566.12001)] which contains complete density results for any abelian extensions of \(\mathbb Q\) and also an account of the arithmetical errors made by A. M. Baily [J. Reine Angew. Math. 315, 190–210 (1980; Zbl 0421.12007)]. From the formulas in section 7 and Theorem 8.4 in Mäki’s paper it is easy to derive an expression for \(N_q(x)\) in the general case, at the cost of a slightly weaker error term.


11R16 Cubic and quartic extensions
11R29 Class numbers, class groups, discriminants
11R45 Density theorems
11R47 Other analytic theory
Full Text: DOI Euclid


[1] Ou, Z. M., and Williams, K. S.: On the density of cyclic quartic fields. Canad. Math. Bull., 44 , 97-104 (2001). · Zbl 0996.11068 · doi:10.4153/CMB-2001-012-6
[2] Spearman, B. K., and Williams, K. S.: Integers which are discriminants of bicyclic or cyclic quartic fields. JP J. Algebra Number Theory Appl., 1 , 179-194 (2001). · Zbl 1038.11068
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.