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.