• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.375-384
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• 2008

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 $1{\leq}i{\leq}{\ell}-1$.