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We study average $2$-torsion in the class group of monogenised fields of odddegree. Bhargava-Hanke-Shankar have recently shown that the average number ofnon-trivial $2$-torsion elements in the class group of monogenised cubic fieldsof a fixed signature is twice the value predicted by theCohen-Lenstra-Martinet-Malle heuristic for the full family. Fix any odd degree$n$ and signature $(r_{1},r_{2})$. In this work, we prove that the averagenumber of non-trivial elements in the class group of monogenised fields ofdegree $n$ and signature $(r_{1},r_{2})$ is bounded by twice the valuepredicted by Cohen-Lenstra-Martinet-Malle. Conditional on a tail estimate for$n \ge 5$, this shows that the doubling phenomenon uncovered for cubic fieldsby Bhargava-Hanke-Shankar persists across all odd degrees and signatures.

Effect of monogenicity on $2$-torsion in the class group of number fields of odd degree