Galois groups of random integer polynomials and van der Waerden's Conjecture

• 成果类型: Article
• 署名作者: Bhargava, Manjul
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2025.201.2.1
• 发表日期: 2025
• 页码: 339-377
• 关键词: primitive permutation-groups; number-fields; upper-bounds; discriminants; density; irreducibility; EXTENSIONS; ORDER; rings

摘要

Of the (2H+1)(n) monic integer polynomials f(x)=x(n)+a1x(n-1)+...+a(n) with max{|a(1)|,...,|a(n)|} <= H, how many have associated Galois group that is not the full symmetric group S-n?There are clearly >> Hn-1 such polynomials, as may be obtained by setting a(n)=0. In 1936, van der Waerden conjectured that O(Hn-1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n <= 4, due to work of van der Waerden and Chow and Dietmann. The purpose of this paper is to prove van der Waerden's Conjecture for all degrees n.