Densities of quartic fields with even Galois groups
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- by Siman Wong
- Proc. Amer. Math. Soc. 133 (2005), 2873-2881
- DOI: https://doi.org/10.1090/S0002-9939-05-07921-9
- Published electronically: April 20, 2005
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Abstract:
Let $N(d, G, X)$ be the number of degree $d$ number fields $K$ with Galois group $G$ and whose discriminant $D_K$ satisfies $|D_K| \le X$. Under standard conjectures in diophantine geometry, we show that $N(4, A_4, X) \ll _\epsilon X^{2/3+\epsilon }$, and that there are $\ll _\epsilon N^{3+\epsilon }$ monic, quartic polynomials with integral coefficients of height $\le N$ whose Galois groups are smaller than $S_4$, confirming a question of Gallagher. Unconditionally we have $N(4, A_4, X) \ll _\epsilon X^{5/6 + \epsilon }$, and that the $2$-class groups of almost all Abelian cubic fields $k$ have size $\ll _\epsilon D_k^{1/3+\epsilon }$. The proofs depend on counting integral points on elliptic fibrations.References
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Bibliographic Information
- Siman Wong
- Affiliation: Department of Mathematics & Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305
- Email: siman@math.umass.edu
- Received by editor(s): March 11, 2004
- Received by editor(s) in revised form: June 7, 2004
- Published electronically: April 20, 2005
- Additional Notes: The author was supported in part by NSA grant H98230-05-1-0069
- Communicated by: Wen-Ching Winnie Li
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 2873-2881
- MSC (2000): Primary 11G05; Secondary 11G35, 11R16, 11R29
- DOI: https://doi.org/10.1090/S0002-9939-05-07921-9
- MathSciNet review: 2159764