Monogenic Fields with Odd Class Number

We prove an upper bound on the average number of $2$-torsion elements in the class group monogenised fields of any degree $n \ge 3$, and, conditional on a widely expected tail estimate, compute this average exactly. As an application, we show that there are infinitely many number fields with odd class number in any even degree and signature. This completes a line of results on class number parity going back to Gauss.Ph.D