Galois Groups of Generic Polynomials

2015/11/19 by Igor Rivin, Rivin, Igor
Computer Science · Mathematics · #11C08 #11R32 #11R45 #Advanced Algebra and Geometry #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1511.06446

openalex publication_date 2015/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the Galois group of a random monic polynomial %of degree d>12 with integer coefficients between -N and N is NOT S_d with probability ≪ \fraclog^Ω(d)NN. Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.

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