Lang-Nagata theorem for Ci fields ($K\text{ is }{C}_i⟹L\text{ is }{C}_{i+s}$)

A finitely generated field extension of transcendence degree $s$ of a $C_i$ field is $C_{i+s}$. In particular a function field of a curve over a C1 field is $C_2$. The theorem includes stability under finite algebraic extensions when $s=0$.