An improved error term for counting D 4 $D_4$ -quartic fields

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-06-17 DOI:10.1112/blms.13106

Kevin J. McGown, Amanda Tucker

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Abstract

We prove that the number of quartic fields $K$ with discriminant $| Δ_{K} | ⩽ X$ whose Galois closure is $D_{4}$ equals $C X + O (X^{5 / 8 + ε})$ , improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting quartic dihedral extensions over an arbitrary base field.

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计算 D 4 $D_4$ 方场的改进误差项

我们证明了具有判别式 | Δ K | ⩽ X $|\Delta _K|\leqslant X$ 且伽罗瓦闭包是 D 4 $D_4$ 的四元数域 K $K$ 等于 C X + O ( X 5 / 8 + ε ) $CX+O(X^{5/8+\varepsilon})$，改进了科恩、迪亚兹和奥利维尔的一个著名结果中的误差项。我们证明了任意基域上的四元二面扩展计数的类似结果。

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