## Algebraic Numbers, An Introduction

### Introduction

As the name suggests,
algebraic number theory employs
modern algebraic techniques to solve problems in number theory.
We are therefore interested in integral extensions of **Z** (the integers),
and field extensions of **Q** (the rationals).
However, other extensions will prove useful as well.
A global field is a finite separable extension of **Q** or of **Z**p(t).
We are extending the rationals,
or the quotients of polynomials in t over the integers mod p.

A number field is a finite extension of **Q**.
Since **Q** has characteristic 0, extensions of **Q** are automatically separable.
Thus a number field is a global field.
In contrast, **Z**p(t)
supports inseparable extensions, such as the pth root of t.
We want to avoid those;
hence a global field is declared separable.

within the context of a global field, an algebraic number is algebraic over the base field,
either **Q** or **Z**p(t).
Of course every element in a finite extension is algebraic,
so everything in a global field qualifies as an algebraic number.

An algebraic integer is integral over the base ring, **Z** or **Z**p[t].
For example, sqrt(2) is an algebraic integer, while 1/3 is not.