Galois Groups of Generic Polynomials

Igor Rivin

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arXiv · 1511.06446

Galois Groups of Generic Polynomials

Igor Rivin

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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 $\ll \frac{\log^{Ω(d)}N}{N}.$ 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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BibTeX RIS

Igor Rivin. 2015-11-19. Galois Groups of Generic Polynomials. https://arxiv.org/abs/1511.06446

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