Irreducible polynomials of bounded height

Abstract

The goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\dots ,210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove that the Galois group of the random polynomial contains the alternating group, again with probability tending to $1$.

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References in corpus (6)

• Irreducibility of random polynomials of large degree
• Permutations fixing a k-set
• Invariable generation of the symmetric group
• Galois Groups of Generic Polynomials
• Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability
• Supplement to the article "irreducible polynomials with bounded height"

Cited by in corpus (10)

• Irreducibility of random polynomials of large degree
• Irreducibility of random polynomials: general measures
• The characteristic polynomial of a random matrix
• Galois groups of random additive polynomials
• Cycle type of random permutations: A toolkit
• Irreducibility of the characteristic polynomials of random tridiagonal matrices
• Distribution of the number of zeros of polynomials over a finite field
• A note on invariable generation of nonsolvable permutation groups
• Irreducibility of random polynomials of $\mathbb{Z}[X]$
• Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem