... Annals of Mathematics, 163 (2006), 723–741 The number of extensions of a number field with fixed degree and bounded discriminant By Jordan S Ellenberg and Akshay Venkatesh* Abstract We give an ... elementary arguments from the geometry of numbers and linear algebra Acknowledgments The authors are grateful for the hospitality of the American Institute of Mathematics, where the first phase of ... in case K = Q, and by Datskovsky and Wright in general [6]; and for n = 4, and K = Q by Bhargava [3], [2] A weaker version of the conjecture for n = was also recently established by Kable and...