Sorry for all the algebra questions everybody, I have my exam on Wednesday (which I'm going to fail) and seem to be struggling with everything as it's so abstract...

Let $\displaystyle g = X^3 +2X - 1 $

Let F be the smallest subfield of R containing a root $\displaystyle \alpha$ of g. What are the elements of F?

Would this just be the identity element 1?

What is the dimension of F considered as a vector space over the rationals?

This I just do not understand at all....can anyone please explain?

Thanks in advance!