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TopicsNumber theory CM fields Root discriminant

The relative class number of a CM field is bounded explicitly in terms of its relative root discriminant and degree (Louboutin's bound).

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Louboutin's 2000 paper in the Journal of Number Theory gives h^-(K) ≤ 2 Q_K w_K sqrt(Δ_K/Δ_F) (e log(Δ_K/Δ_F) / (4π d))^d for a CM field K of degree 2d over its maximal real subfield F, where Q_K is the Hasse unit index and w_K the number of roots of unity. It is a refereed result that has been used for a quarter century in class number computations for CM fields and is not disputed.

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