0Introduction

III Local Fields

0 Introduction
What are local fields? Suppose we are interested in some basic number theoretic
problem. Say we have a polynomial
f
(
x
1
, ··· , x
n
)
Z
[
x
1
, ··· , x
n
]. We want to
look for solutions
a Z
n
, or show that there are no solutions at all. We might
try to view this polynomial as a real polynomial, look at its roots, and see if
they are integers. In lucky cases, we might be able to show that there are no
real solutions at all, and conclude that there cannot be any solutions at all.
On the other hand, we can try to look at it modulo some prime
p
. If there
are no solutions mod
p
, then there cannot be any solution. But sometimes
p
is
not enough. We might want to look at it mod
p
2
, or
p
3
, or . . . . One important
application of local fields is that we can package all these information together.
In this course, we are not going to study the number theoretic problems, but
just look at the properties of the local fields for their own sake.
Throughout this course, all rings will be commutative with unity, unless
otherwise specified.