## Upper bounds for the number of number fields with alternating Galois group

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- by Eric Larson and Larry Rolen PDF
- Proc. Amer. Math. Soc.
**141**(2013), 499-503 Request permission

## Abstract:

Let $N(n, A_n, X)$ be the number of number fields of degree $n$ whose Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(n, A_n, X)\sim C_n\cdot X^{\frac {1}{2}} \cdot (\log X)^{b_n}$ for constants $b_n$ and $C_n$. For $6 \leq n \leq 84393$, the best known upper bound is $N(n, A_n, X) \ll X^{\frac {n + 2}{4}}$, by Schmidt’s theorem, which implies there are $\ll X^{\frac {n + 2}{4}}$ number fields of degree $n$. (For $n > 84393$, there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that $N(n, A_n, X) \ll X^{\frac {n^2 - 2}{4(n - 1)}+\epsilon }$, thereby improving the best previous exponent by approximately $\frac {1}{4}$ for $6 \leq n \leq 84393$.## References

- Manjul Bhargava,
*The density of discriminants of quartic rings and fields*, Ann. of Math. (2)**162**(2005), no. 2, 1031–1063. MR**2183288**, DOI 10.4007/annals.2005.162.1031 - Manjul Bhargava,
*The density of discriminants of quintic rings and fields*, Ann. of Math. (2)**172**(2010), no. 3, 1559–1591. MR**2745272**, DOI 10.4007/annals.2010.172.1559 - Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier,
*A survey of discriminant counting*, Algorithmic number theory (Sydney, 2002) Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 80–94. MR**2041075**, DOI 10.1007/3-540-45455-1_{7} - H. Davenport and H. Heilbronn,
*On the density of discriminants of cubic fields. II*, Proc. Roy. Soc. London Ser. A**322**(1971), no. 1551, 405–420. MR**491593**, DOI 10.1098/rspa.1971.0075 - Jordan S. Ellenberg and Akshay Venkatesh,
*The number of extensions of a number field with fixed degree and bounded discriminant*, Ann. of Math. (2)**163**(2006), no. 2, 723–741. MR**2199231**, DOI 10.4007/annals.2006.163.723 - Gunter Malle,
*On the distribution of Galois groups*, J. Number Theory**92**(2002), no. 2, 315–329. MR**1884706**, DOI 10.1006/jnth.2001.2713 - J. Pila,
*Density of integer points on plane algebraic curves*, Internat. Math. Res. Notices**18**(1996), 903–912. MR**1420555**, DOI 10.1155/S1073792896000554 - Wolfgang M. Schmidt,
*Number fields of given degree and bounded discriminant*, Astérisque**228**(1995), 4, 189–195. Columbia University Number Theory Seminar (New York, 1992). MR**1330934** - David J. Wright,
*Distribution of discriminants of abelian extensions*, Proc. London Math. Soc. (3)**58**(1989), no. 1, 17–50. MR**969545**, DOI 10.1112/plms/s3-58.1.17

## Additional Information

**Eric Larson**- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- Email: elarson3@gmail.com
**Larry Rolen**- Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- Email: larry.rolen@mathcs.emory.edu
- Received by editor(s): July 6, 2011
- Published electronically: June 25, 2012
- Additional Notes: The authors are grateful for the support of the NSF in funding the Emory 2011 REU. The authors would like to thank our advisor, Andrew Yang, as well as Ken Ono for their guidance, useful conversations, improving the quality of exposition of this article, and hosting the REU
- Communicated by: Matthew A. Papanikolas
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 499-503 - MSC (2010): Primary 11R99, 11G99
- DOI: https://doi.org/10.1090/S0002-9939-2012-11543-6
- MathSciNet review: 2996953