Let $k={mathbb{F}}_q(T)$ and ${mathbb{A}}={mathbb{F}}_q[T]$. Fix a prime divisor ${ell}$ q-1. In this paper, we consider a ${ell}$-cyclic real function field $k(sqrt[{ell}]P)$ as a subfield of the imaginary bicyclic function field K = $k(sqrt[{ell}]P,;(sqrt[{ell}]{-Q})$, which is a composite field of $k(sqrt[{ell}]P)$ wit a ${ell}$-cyclic totally imaginary function field $k(sqrt[{ell}]{-Q})$ of class number one. und give various conditions for the class number of $k(sqrt[{ell}]{P})$ to be one by using invariants of the relatively cyclic unramified extensions $K/F_i$ over ${ell}$-cyclic totally imaginary function field $F_i=k(sqrt[{ell}]{-P^iQ})$ for $1{leq}i{leq}{ell}-1$