Cyclotomic Extensions, An Introduction

Introduction

Before we proceed, you need to be familiar with cyclotomic extensions in general.

As you might guess, a cyclotomic number field is a number field that is also a cyclotomic extension. Adjoin y, the n^th root of 1. Remember that y is a root of ζ_n(x), which has degree φ(n). Gauss proved this polynomial is irreducible over Q, and over Z. The conjugates of y are the powers of y, where the exponent is coprime to n.

Localization

Since gauss' lemma applies, a polynomial p is irreducible over Z iff it is irreducible over Q. Let K be a ring of fractions between Z and Q. If p splits over Z it splits over K, and it splits over Q, and it splits over Z. Thus p is irreducible over Z iff it is irreducible over K. Apply this to ζ, and ζ remains irreducible under any localization of Z.

Global Field

A global field could be a number field based on Q, or it could be an extension of K, where K = Z_p(t). Adjoin y to K, and the extension is the same as adjoining y to Z_p, and then adjoining the indeterminant t. Note that t has not suddenly become algebraic over Z_p(y), else it would be algebraic over Z_p. Therefore the cyclotomic global field is the extension of Z_p by y, giving another finite field, then t is adjoined.