Extension Field as Vector Space

Theorem

Given an extension field K of F, K is a vector space over the scalar field 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 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 K, that is, let u,vK and α,βF, then α,βK since FK, and trivially from the properties of the field F, we have that