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

2022/04/04 by Manjul Bhargava, Arul Shankar, Bhargava, Manjul +3 · 2 citations
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.01331

openalex publication_date 2022/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≤ n ≤ 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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• Improved bounds on number fields of small degree
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• Discriminants of Fields Generated by Polynomials of Given Height
• The number of extensions of a number field with fixed degree and bounded discriminant