Prove that the field F is a vector space over itself.

How can I prove that a field F is a vector space over itself?
Intuitively, it seems obvious because the definition of a field is
nearly the same as that of a vector space, just with scalers instead
of vectors.

Here’s what I’m thinking:
Let V={(a)|a in F} describe the vector space for F. Then I just show
that vector addition is commutative, associative, has an identity and
an inverse, and that scalar multiplication is distributaries,
associative, and has an identity.

Example 1: commutativity of addition
x,y in V
(x)+(y)=(y)+(x)
(x+y)=(y+x) vector addition
(X+y)=(x+y) associative property

Example 2: additive inverse
x,y,0 in V
(x)+(y)=(0)
(x+y)=(0) vector addition
Let y=-x in V
(X+-x)=(0) substitute
(0)=(0) simplify

I don’t know if I’m going in the right direction with this, although
it seems like it should be a pretty simple proof. I think mostly I’m
having trouble with the notation.

Any help would be greatly appreciated! Thanks in advance!

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