On real quadratic function fields of Chowla type with ideal class number one

Abstract

Let F q \mathbb {F}_q be the finite field with q q elements, (2 ⧸ | q {\not |}q ), k = F q ( x ) k= \mathbb {F}_q(x) , K = k ( D ) K=k(\sqrt {D}) where D = D ( x ) = A ( x ) 2 + a D=D(x) =A(x)^2+a is a square-free polynomial in F q [ x ] \mathbb {F}_q[x] with deg ⁡ A ( x ) ≥ 1 \deg A(x)\geq 1 and a ∈ F q ∗ a\in \mathbb { F}_q^* . In this paper several equivalent conditions for the ideal class number h ( O K ) h(O_K) to be one are presented and all such quadratic function fields with h ( O K ) = 1 h(O_K)=1 are determined.