## View abstract

### Session S09 - Number Theory in the Americas

Thursday, July 15, 11:00 ~ 12:00 UTC-3

## Galois groups of random integer polynomials

### Manjul Bhargava

#### Princeton , USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka8452a3e9785b3be97a97435559255f7').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya8452a3e9785b3be97a97435559255f7 = 'bh&#97;rg&#97;v&#97;' + '&#64;'; addya8452a3e9785b3be97a97435559255f7 = addya8452a3e9785b3be97a97435559255f7 + 'm&#97;th' + '&#46;' + 'pr&#105;nc&#101;t&#111;n' + '&#46;' + '&#101;d&#117;'; var addy_texta8452a3e9785b3be97a97435559255f7 = 'bh&#97;rg&#97;v&#97;' + '&#64;' + 'm&#97;th' + '&#46;' + 'pr&#105;nc&#101;t&#111;n' + '&#46;' + '&#101;d&#117;';document.getElementById('cloaka8452a3e9785b3be97a97435559255f7').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya8452a3e9785b3be97a97435559255f7 + '\'>'+addy_texta8452a3e9785b3be97a97435559255f7+'<\/a>';

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as can be seen by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known for $n\leq 4$ due to work of van der Waerden and Chow and Dietmann. In this talk, we describe how to prove van der Waerden's Conjecture for all degrees $n$.