Fulton, Algebraic Curves, Exercise 1.37 I am stuck trying to prove this:


  Let $k$ be any field, $F\in k[X]$ a monic polynomial of degree $n&gt;0$.
  Then the residues $\bar{1},\bar{X},...,\overline{X^{n-1}}$ form a
  basis for $k[X]/(F)$ over $k$.


I thought I had done it correctly but my proof isn't convincing me, so back to square one.

Any help will be greatly appreciated!

EDIT: The question does say "any field" but it is in the section that says $k$ is algebraically closed for the whole section. 

Here's the attempt.

Let $G$ be any polynomial with residue $\bar{G}\in k[X]/(F)$, so $G-aF^m=0$ for some $a\in k$ and $m&gt;0$. Since $k$ is algebraically closed, we may divide $G$ by the irreducible factors of $F$ so that $G$ does not vanish for any $P\in V(F)$. Moreover, since $F$ is monic, $\deg(G)=\deg(G-aF^m)+1$ for some $a\in k$ and $m&gt;0$. It follows then that $\{\bar{1},\bar{X},...,\overline{X^{n-1}}\}$ is a basis for $k[X]/(F)$.
The fact that $F$ is monic is useful it says that 
$$x^{n}=a_1x^{n-1}+\cdots +a_n.$$
The fact that $F$ is monic is useful it says that 
$$x^{n}=a_1x^{n-1}+\cdots +a_n.$$