The extended Bloch groups of biquadratic and dihedral number fields

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index Q2(F) in Browkin and Gangl's formulas on the Brauer-Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for Q2(F)=1 or 2 in terms of the Tate kernels. We also prove that Q2(F)=1 or p for any dihedral extension F/Q whose Galois group is the dihedral group of order 2p, where p is an odd prime.