Geodesic
The number of extensions of a number field with fixed degree and bounded discriminant
Jordan S. Ellenberg1 ; Akshay Venkatesh2
1Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, United States
2Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States
Annals of mathematics, Tome 163 (2006) no. 2, pp. 723-741
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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.

DOI : 10.4007/annals.2006.163.723

Jordan S. Ellenberg  1   ; Akshay Venkatesh  2

1 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, United States
2 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States 
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Jordan S. Ellenberg; Akshay Venkatesh. The number of extensions of a number field with fixed degree and bounded discriminant. Annals of mathematics, Tome 163 (2006) no. 2, pp. 723-741. doi: 10.4007/annals.2006.163.723

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