Using notations to write in unique form Let $I=(X^3+2)$ be the principal ideal of $\mathbb{F}_7[X]$ generated by $X^3+2$. Use the notation $a=a+I$ for a in $\mathbb{F}_7$ and alpha=X+I. Show that any element of $\mathbb{F}_7[X]/I$ can be written uniquely in the form $a+b*alpha+c*alpha^2$ for a,b,c in $\mathbb{F}_7$ and that $alpha^3=5$.
I have no idea how to do this. Please help.
 A: We can use the following fact:
Question: Let $F$ be a field, and let $p(x) \in F[x]$ be a polynomial of degree $n \geq 1$. Let bars denote passage to $F[x]/(p(x))$. Prove that for every $q(x) \in F[x]$ there is a unique $q_0(x) \in F[x]$ such that $q_0(x)$ has degree less than $n$ and $\overline{q(x)} = \overline{q_0(x)}$.
Solution: Since $F$ is a field, $F[x]$ is an Euclidean Domain. So there exists $a(x), q_0(x) \in F[x]$ such that $q(x) = a(x)p(x) + q_0(x)$ where deg$q_0(x) \leq n-1$. Clearly, $\overline{q(x)} = \overline{q_0(x)}$. The uniqueness of $q_0(x)$ follows from the proof that $F[x]$ is an Euclidean Domain.
So in particular, the elements $\overline{1}, \overline{x}, \cdots, \overline{x^{n-1}}$ form a basis of the vector space $F[x]/(p(x))$ over $F$.
Now in the given situation, $1, \alpha, \alpha^2$ forms a basis for the vactor space $\mathbb{F}_7[X]/I$ over $\mathbb{F}_7$. Hence the uniqueness of the form $a + b\alpha +c\alpha^2$. Also note that, in the ring $\mathbb{F}_7[X]/I$, $\alpha^3 + 2 = 0$, i.e. $\alpha^3 = -2 = 5$.
