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Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class number $N$? ($U_E$ denotes the group of units of $E$.)

(Indeed, I think if $E/F$ is an abelian (or at least is a cyclic) unramified (at all finite places) extension, it is not possible for $\# H^1(G,U_E)$ to be relatively prime to the class number $N$. Is that true?)

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If I understand your question correctly, every simple unramified extension of a quadratic number field with class number $1$ is an example. Artin constructed the first such extensions, now there are many examples known; see e.g. "Remark on infinite unramified extensions of number fields with class number one" by D. Brink and the literature there.

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