The Picard group { \Gamma} is PSL(2,Z[i]). We have defined coset diagram for the Picard group. It has been observed that some elements of Q(i,/surd3) of the form ((a+b/surd3)/c) and their conjugates ((a-b/surd3)/c) over \mathbb{ Q}(i) have different signs in the coset diagram for the action of { \Gamma} on the biquadratic field Q(i,/surd3), these are called ambiguous numbers. We have noticed that ambiguous numbers in the coset diagram for the action of { \Gamma} on \mathbb{ Q}(i,/surd3) form a unique pattern. It has been shown that there are finite number of ambiguous numbers in an orbit { \Gamma}{ \alpha}, where { \alpha} is ambiguous, and they form a closed path and it is the only closed path in the orbit { \Gamma}{ \alpha}. We have devised a procedure to obtain ambiguous numbers of the form ((a+k/surd3)/c), where k is a positive integer.
Categories: math.GR