Under the assumption of the generalized Riemann Hyothesis verifying the class number belongs to NPcapCo-NP

We show that under the assumption of a certain Generalized Riemann Hypothesis the problem of verifying the value of the class number of an arbitrary algebraic number field F of arbitrary degree belongs to the complexity class NP ∩ co-NP. In order to prove this result we introduce a compact representation of algebraic integers which allows us to represent a system of fundamental units by (2 + log2(Δ))O(1) bits, where Δ is the discriminant of F.

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