Random polynomials, probabilistic Galois theory, and finite field arithmetic

Abstract: In the talk we will discuss recent advances on the following two questions: Let $A(X) = \sum ±X^i$ be a random polynomial of degree n with coefficients taking the values $-1, 1$ independently each with probability $1/2$.

Q1: What is the probability that $A$ is irreducible as the degree goes to infinity?

Q2: What is the typical Galois group of $A$?

One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problems and partial results.

number theory

Audience: researchers in the topic

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Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

We will email out the link to all registered participants the day before.

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