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 Q 2 ( 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 Q 2 ( F ) = 1 or 2 in terms of the Tate kernels. We also prove that Q 2 ( F ) = 1 or p for any dihedral extension F / Q whose Galois group is the dihedral group of order 2 p , where p is an odd prime. [ABSTRACT FROM AUTHOR]