Certain Quadratic Unramified Extensions of F(T, √D (T)) and Theory of Genera for Quadratic Extensions of Fq(T)

Let F be a field of characteristic ≠ 2, k₀ = F(T), and let k be a quadratic extension of k₀. We determine explicitly (i) certain quadratic unramified extensions of k, (ii) the Geschlechterkdrper of k and (iii) the real characters of the ideal class group of the integral closure of F[T] in k.