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 distributary, associative, and has an identity.

Example 1: Commutativity of addition, Here x,y are in V

(x)+(y) = (y)+(x)

(x+y)=(y+x): vector addition

x + (X+y) = (X + 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!

Solution 1:

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.