Part I
Basic Theory

Example. $f (x_{1}, \dots, x_{r}) \in ℤ [c_{1}, \dots, x_{r}]$ , $f (x_{1}, \dots, x_{r}) = 0$ ? This is hard to study. It is easier to study

\begin{array}{l} f (x_{1}, \dots, x_{r}) & \equiv 0 (m o d p) \\ f (x_{1}, \dots, x_{r}) & \equiv 0 (m o d p^{2}) \\ ⋮ \\ f (x_{1}, \dots, x_{r}) & \equiv 0 (m o d p^{n}) \end{array}

A local field packages all this information together.

p

Part IBasic Theory