Algebraic Numbers, An Introduction

Introduction

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 p^th 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.