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

Manjul Bhargava; Arul Shankar; Xiaoheng Wang

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arXiv · 2204.01331

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

Manjul Bhargava·Arul Shankar·Xiaoheng Wang

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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 $6 \leq n \leq 94$. 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.

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Manjul Bhargava, Arul Shankar, Xiaoheng Wang. 2022-04-04. An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree. https://arxiv.org/abs/2204.01331

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