A proof of van der Waerden’s Conjecture on random Galois groups of polynomials

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Abstract

Of the (2H +1)n monic integer polynomials f(x) = xn+a1xn−1+···+an with max{|a1|,…,|an|} ≤ H,howmanyhave associated Galois group that is not the full symmetric group Sn? There are clearly ≫ Hn−1 such polynomials, as may be obtained by setting an = 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. In this expository article, we outline a proof of van der Waerden’s Conjecture for all degrees n..

Original languageAmerican English
Pages (from-to)45-60
Number of pages16
JournalPure and Applied Mathematics Quarterly
Volume19
Issue number1
DOIs
StatePublished - 2023

ASJC Scopus subject areas

  • General Mathematics

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