The number of extensions of a number field with fixed degree and bounded discriminant


We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant $\leq X$; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions.


Jordan S. Ellenberg

Department of Mathematics
University of Wisconsin-Madison
Madison, WI 53706
United States

Akshay Venkatesh

Courant Institute of Mathematical Sciences
New York University
New York, NY 10012
United States