Number fields and number rings

Abstract

A number field is a subfield of ℂ having finite degree (dimension as a vector space) over ℚ. We know (see appendix 2) that every such field has the form ℚ[α] for some algebraic number α ∈ ℂ. If α is a root of an irreducible polynomial over ℚ, having degree n, then

$$\mathbb{Q}[\alpha ] = \left\{ {{a_o} + {a_1}\alpha + \cdots + {a_{n - 1}}{\alpha ^{n - 1}}:{a_1} \in \mathbb{Q}{\kern 1pt} \forall i} \right\}$$

and representation in this form is unique; in other words, {1, α,…,α^n-1} is a basis for ℚ[α] as a vector space over ℚ.