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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.

MSC:

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

[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
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