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

By Jordan S. Ellenberg, Akshay Venkatesh. 2003

114 citations recorded by the source corpus.

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant ≤ 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/62b25ecf13d1d7af61f6320f70a7ede5727f45a4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Counting primitive points of bounded height (2010, unknown): “The latter is proved only for very special cases although great progress was achieved by the recent work of Ellenberg and Venkatesh [4].”
Counting extensions of function fields with bounded discriminant and specified Galois group (2005, methodology): “Counting extensions of function fields with bounded discriminant and specified Galois group Jordan S. Ellenberg and Akshay Venkatesh∗ 29 Apr 2004”
A construction of polynomials with squarefree discriminants (2011, methodology): “This is far inferior to what can be shown using other methods: for n ≤ 5 the number of such fields is known to be O(N) by work of Bhargava [2, 3], while for larger n, Ellenberg and Venkatesh [5] have given a lower bound of cN 2 .”
Squarefree values of polynomial discriminants I (2016, background): “Following [9], for any algebraic number x, we write ‖x‖ for the maximum of the archimedean absolute values of x.”