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
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.
$\blacksquare$