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### Session S09 - Number Theory in the Americas

Thursday, July 15, 13:00 ~ 13:30 UTC-3

## Irreducibility of random polynomials of large degree

### Dimitris Koukoulopoulos

#### U. Montreal, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak919503db01408f15575bd46b20e49c6a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy919503db01408f15575bd46b20e49c6a = 'd&#105;m&#105;tr&#105;s.k&#111;&#117;k&#111;&#117;l&#111;p&#111;&#117;l&#111;s' + '&#64;'; addy919503db01408f15575bd46b20e49c6a = addy919503db01408f15575bd46b20e49c6a + '&#117;m&#111;ntr&#101;&#97;l' + '&#46;' + 'c&#97;'; var addy_text919503db01408f15575bd46b20e49c6a = 'd&#105;m&#105;tr&#105;s.k&#111;&#117;k&#111;&#117;l&#111;p&#111;&#117;l&#111;s' + '&#64;' + '&#117;m&#111;ntr&#101;&#97;l' + '&#46;' + 'c&#97;';document.getElementById('cloak919503db01408f15575bd46b20e49c6a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy919503db01408f15575bd46b20e49c6a + '\'>'+addy_text919503db01408f15575bd46b20e49c6a+'<\/a>';

Let $f(x)=a_0+a_1x+\cdots+a_{n-1}x^{n-1}+x^n$ be a random monic polynomial, where $a_j$ is chosen uniformly at random from $\{0,1\}$ and independently of the other coefficients. In 1993, Odlyzko and Poonen conjectured that $f(x)$ is irreducible with probability $\sim1/2$ when $n\to\infty$. Breuillard and Varj\'u proved that this expectation is indeed true under the Generalized Riemann Hypothesis. In this talk, I will present recent joint work with Bary-Soroker and Kozma that proves that $f(x)$ is irreducible with probability $\ge1/1000$ for all large enough $n$. In addition, if we condition on the event that $f(x)$ is irreducible, then we prove that the Galois group of $f(x)$ contains the alternating group $A_n$ with conditional probability $\sim1$. The proofs use a fun mixture of ideas from sieve methods, the arithmetic of polynomials over finite fields, $p$-adic Fourier analysis, primes with restricted digits, Galois theory and group theory.