# 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 + (X+y) = (X + x)+y: associative property

Example 2: Additive inverse x,y,0 in V (x)+(y)=(0)

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!

• Yes, your notation is the problem. Just drop the brackets. "vector" and "scalar" are merely terminology, and can be ignored in this case. Sep 14 '13 at 17:26
• Note that $\mathbb{F}$ is a vector space over itself, replacing scalar multiplication with field multiplication. Sep 14 '13 at 23:01