Part I  Basic Theory  Example  f   x 1       x r         c 1       x r    f   x 1       x r     0  This is hard to study  It is easier to study     f   x 1       x r     0   m o d p   f   x 1       x r     0   m o d p 2     f   x 1       x r     0   m o d p n       A local field packages all this information together    1 Absolute values  2 Valuation Rings  3 The p adic numbers