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arXiv · 2204.02836

A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields

Erich L. Kaltofen
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Abstract

Let f(x) = x^n + (a[n-1] t + b[n-1]) x^(n-1) + ... + (a[0] t + b[0]) be of constant degree n in x and degree <= 1 in t, where all a[i],b[i] are randomly and uniformly selected from a finite field GF(q) of q elements. Then the probability that the Galois group of f over the rational function field GF(q)(t) is the symmetric group S(n) on n elements is 1 - O(1/q). Furthermore, the probability that the Galois group of f(x) over GF(q)(t) is not S(n) is >= 1/q for n >= 3 and > 1/q - 1/(2q^2) for n = 2.

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Erich L. Kaltofen. 2022-04-06. A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields. https://arxiv.org/abs/2204.02836

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