A prime decomposition symbol for a non-abelian central extension which is abelian over a bicyclic biquadratic field

Abstract

In a previous paper [6] we had some criteria for the prime decomposition in certain non-abelian extensions over the rational number field Q, and as its special case we had a reciprocity of the biquadratic residue symbol. The reciprocity was obtained by using a descent method of the prime decomposition for a central extension over Q which is abelian over a biquadratic field In the present paper we study on the case over a biquadratic field in general. We define a symbol [d1, d2, p] which expresses the decomposition law of a rational prime p in a central extension mentioned above.