Global field

Remarks

Example: $\Q(\sqrt{2})$ is a global field, because it has degree $2$ over $\Q$. Example: $\F_3(t)$ itself is a global function field, of degree $1$ over $\F_3(t)$. Non-example: $\R$ is not a global field, because it has infinite degree over $\Q$ and has characteristic $0$. Non-example: $\C(t)$ is not a global field, because its field of constants $\C$ is not finite. The global function fields are exactly the function fields of smooth projective curves over finite fields.