Extension Field as Vector Space

Theorem

Given an extension field \(\mathbb{K}\) of \(\mathbb{F}\), \(\mathbb{K}\) is a vector space over the scalar field \(\mathbb{F}\).

This result is extremely simple to prove, however it is very useful in the theory of field extensions, such as through explicitly finding a basis for an extension field. It also allows us to talk about the dimension of the field as a vector space.

Proof

Clearly \(\mathbb{K}\) is an abelian group under addition, from the definition of a field.

Then, all other field axioms hold because their equivalents hold for elements in \(\mathbb{K}\), that is, let \(u, v \in \mathbb{K}\) and \(\alpha, \beta \in \mathbb{F}\), then \(\alpha, \beta \in \mathbb{K}\) since \(\mathbb{F} \subseteq \mathbb{K}\), and trivially from the properties of the field \(F\), we have that