Tractability of Algebraic Function Fields in One Variable of Genus Zero Over Global Fields

If the condition described in Definition 1.1 on the simultaneous representation of quaternion algebras holds over a field F, then F is called tractable. In this paper, we show that a field F is tractable if and only if any purely transcendental extension of F is tractable. Next, if F is a global field of characteristic unequal to two, we consider tractability of an algebraic function field K in one variable of genus zero over F and determine precisely when K is tractable if F has at most one ordering.