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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

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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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If in the axioms of vector spaces you assume that the vector space is the same as the field, and you identify vector addition and scalar multiplication respectively with addition and multiplication in the field, you will see that all axioms are contained in the set of axioms of a field. There is nothing more to check than this.

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