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Let k be a number field and K/k Galois. We transform the construction of theunramified Brauer group of the norm one torus R^1_K/k(G_m) into theconstruction of a special abelian extension over K. If k=Q and K/Q biquadratic,we explicitly construct the unramified Brauer group of R^1_K/Q(G_m).

The unramified Brauer group of norm one tori