An improved error term for counting $D_4$-quartic fields
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929334983327744 |
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| author | McGown, Kevin J. Tucker, Amanda |
| author_facet | McGown, Kevin J. Tucker, Amanda |
| contents | We prove that the number of quartic fields $K$ with discriminant $|Δ_K|\leq X$ whose Galois closure is $D_4$ equals $CX+O(X^{5/8+\varepsilon})$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14564 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An improved error term for counting $D_4$-quartic fields McGown, Kevin J. Tucker, Amanda Number Theory 11N45, 11R16 (Primary) 11R29, 11R42, 11R45 (Secondary) We prove that the number of quartic fields $K$ with discriminant $|Δ_K|\leq X$ whose Galois closure is $D_4$ equals $CX+O(X^{5/8+\varepsilon})$, 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. |
| title | An improved error term for counting $D_4$-quartic fields |
| topic | Number Theory 11N45, 11R16 (Primary) 11R29, 11R42, 11R45 (Secondary) |
| url | https://arxiv.org/abs/2307.14564 |