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Monogenic fields with odd class number Part II: even degree
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abstract
In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisely conditional on a tail estimate) using a parametrisation of Wood. These averages are the first $p$-torsion averages to be calculated for $p$ not coprime to the degree (in degree at least $3$), shedding light on the question of Cohen-Lenstra-Martinet-Malle type heuristics for class groups and narrow class groups at "bad" primes.
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Cited by 1 Pith paper
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Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII
The main theorem is false: for φ_{p,c}(z)=z^p+c over F_p with p|c, every point is fixed, so the number of 1_n-preperiodic points is 0, not p.
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