Imaginary bicyclic function fields with the real cyclic subfield of class number one

Let k = F(q)(T) and A = F(q)[T]. Fix a prime divisor l of q - 1. In this paper, we consider a l-cyclic real function field k((l)root P) as a subfield of the imaginary bicyclic function field K = k((l)root P, (l)root-Q), which is a composite field of k((l)root P) with a l-cyclic totally imaginary function field k((l)root-Q) of class number one, and give various conditions for the class number of k((l)root P) to be one by using invariants of the relatively cyclic unramified extensions K/Fi over l-cyclic totally imaginary function field F(i) = k((l)root-P(i)Q) for 1 <= i <= l - 1.