Proving Vector Space of F[x]/(g(x)) with Degree n

[johnson123]

Homework Statement

Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x],
and g has degree n.

Its clear that F[x]/( g(x) ) is a vector space and that

B= (1,x^{2},...,x^{n-1}) spans F[x]/( g(x) ),

but I am having trouble showing that B is linearly independent

I realize this is pretty much a HW problem and it should be in the HW section, but I
read a post from one of the pf mentors noting that for gradlevel/seniorlevel problems
you might have a chance at a response from the non homework sections. thanks for any suggestions.

 

[Hurkyl]

Well, what happens if they are linearly dependent, so that a nontrivial linear combination of them is equal to zero in F[x] / (g(x))?

 

[ejungkurth]

It's not clear that you have tied B to either F[x] or g(x). First relate B to F and g. Assume for the moment that I am not the person who doesn't have the answer.