# Field Theory

1. Feb 7, 2008

### johnson123

1. The problem statement, all variables and given/known data

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 im 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 hw sections. thanks for any suggestions.

2. Feb 7, 2008

### Hurkyl

Staff Emeritus
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))?

3. Feb 7, 2008

### 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.