Extension Field as Vector Space

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

• $\alpha u\in \mathbb{K}$
• $\alpha (\beta v)=(\alpha \beta )v$
• $1v=v$
• $\alpha (u+v)=\alpha u+\alpha v$
• $(\alpha +\beta )u=\alpha u+\beta u$