  Let $p$ be a prime integer and $F$ the function field in two algebraically
independent variables over a smaller field $F_0$. We prove that if
$\operatorname{char}(F_0)=p\geq 3$ then there exist $p^2-1$ cyclic algebras of
degree $p$ over $F$ that have no maximal subfield in common, and if
$\operatorname{char}(F_0)=0$ then there exist $p^2$ cyclic algebras of degree
$p$ over $F$ that have no maximal subfield in common.
