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Is it true that a field is a vector space over a field?

This idea arises in me after reading the solution for the question the order of finite field is $p^n$.

Order of finite fields is $p^n$

I am wondering if it is possible to not consider subfield but just a field as a vector space over a field.

In fact, I don't quite get why a field is a vector space over its subfield. Why can't it be other fields?

I am having a lot of confusion here.

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If $F\subset E$ are fields, then $E$ is an $F$-vector field with the given addtion (in $E$) as addition of vecors and the given multiplication (of element of $F$ (which are also elements of $E$) with other elements of $E$) as scalar multiplication. The vector space axioms follow striaghtforwardly from the field axioms, thus $E$ is a vector space over $F$.

Can this work if $F$ is not a subfield? If we want the addition/multiplication obtainable naturally from the given operations in $E$ and $F$, we must have a vector $f\cdot e$ for each scalar $f\in F$ and vector $e\in E$. Specifically, we can take $e=1$ and - whatever niche multiplication the dot represents - we then have $f=f\cdot 1\in E$.

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