Vector Space on Field Extension is Vector Space

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Theorem

Let $\struct {K, +, \times}$ be a field.

Let $L / K$ be a field extension over $K$.

Let $\struct {L, +, \times}_K$ be the a vector space of $L$ over $K$.


Then $\struct {L, +, \times}_K$ is a vector space.


Proof

We have that $L$ is a

By definition, $L / K$ is a field extension over $K$.

Thus, by definition, $K$ is a subfield of $L$.

Thus, also by definition, $K$ is a division subring of $L$.

The result follows by Vector Space over Division Subring is Vector Space.

$\blacksquare$