@article{a3db2192a1854e3b93e2765eb2c3e44e,

title = "An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree",

abstract = "We prove an improvement on Schmidt's upper bound on the number of number fields of degree n and absolute discriminant less than X for 6 ≤ n ≤ 94. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].",

author = "Manjul Bhargava and Arul Shankar and Xiaoheng Wang",

note = "Funding Information: The first-named author was supported by a Simons Investigator Grant and NSF Grant DMS-1001828. The second-named author was supported by an NSERC Discovery Grant and Sloan Research Fellowship. The third-named author was supported by an NSERC Discovery Grant. We are very grateful to Ashvin Swaminathan and Sameera Vemulapalli for helpful conversations and comments on an earlier version of this manuscript. We thank the referee for many helpful comments and, in particular, for pointing out an issue in the previous proof of what is now Proposition . We also thank Theresa Anderson, Ayla Gafni, Kevin Hughes, Robert Lemke Oliver, David Lowry-Duda, Frank Thorne, Jiuya Wang and Ruixiang Zhang for sharing with us their recent preprint []. Publisher Copyright: {\textcopyright} ",

year = "2022",

month = oct,

day = "10",

doi = "10.1017/fms.2022.71",

language = "English (US)",

volume = "10",

journal = "Forum of Mathematics, Sigma",

issn = "2050-5094",

publisher = "Cambridge University Press",

}