Towards van der Waerden's conjecture
Abstract
How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in $[-H,H]$, is $O(H^{3.91})$. More generally, we show that if $n \ge 3$ and $n \notin \{ 7, 8, 10 \}$ then there are $O(H^{n-1.017})$ monic, irreducible polynomials of degree $n$ with integer coefficients in $[-H,H]$ and Galois group not containing $A_n$. Save for the alternating group and degrees $7,8,10$, this establishes a 1936 conjecture of van der Waerden.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.14593
- arXiv:
- arXiv:2106.14593
- Bibcode:
- 2021arXiv210614593C
- Keywords:
-
- Mathematics - Number Theory;
- 11R32 (primary);
- 11C08;
- 11D45;
- 11G35 (secondary)
- E-Print:
- Incorporated referee suggestions