How to show elements of a quotient ring can be written in a certain way? Let $I=(x^3+2)$ show that all elements of $\mathbb{F}_7[X]/I$ can be written in the form $a+bX+cX^2$ for some $a,b,c \in \mathbb{F_7}.$
I know you normally want to see what I hve attempted but I'm struggling to even begin this question to be honest. I wrote out a coset let $x \in \mathbb{F_7}[X].$ Then $x+I=\{x+y~:y \in I\}$ and I know that $\mathbb{F}_7[X]/I$  is the set of these cosets for all values $x \in  \mathbb{F_7}[X]$.
But I am incredibly stuck any help?
 A: If $R$ is any commutative ring and $f(X)$ is any monic polynomial in $R[X]$, then an element of the quotient ring $R/I$, where $I=(f(X))$ can be written as
$$
a_0+a_1X+\dots+a_{n-1}X^{n-1}+I
$$
where $n$ is the degree of $f(X)$. Just recall that $p(X)\in R[X]$ can be written as
$$
p(X)=f(X)q(X)+r(X)
$$
where the degree of $r(X)$ is less than the degree of $f(X)$, due to the fact that $f(X)$ is monic. The proof is by easy induction on the degree of $p$. (Here I use the convention that the degree of $0$ is $-\infty$; add “or $r(X)=0$” if you don't agree with this convention.)
Now, suppose
$$
a_0+a_1X+\dots+a_{n-1}X^{n-1}+I=b_0+b_1X+\dots+b_{n-1}X^{n-1}+I
$$
Then, setting
$$
d(X)=(a_0-b_0)+(a_1-b_1)X+\dots+(a_{n-1}-b_{n-1})X^{n-1}
$$
we have $d(X)\in I$, so $d(X)=f(X)q(X)$, for some polynomial $q$. But, since $f(X)$ is monic, as soon as $q(X)\ne0$ the degree of $f(X)q(X)$ is the sum of the degrees of $f(X)$ and $q(X)$, so at least $n$. This is a contradiction, unless $d(X)=0$.
So the representation $a_0+a_1X+\dots+a_{n-1}X^{n-1}+I$ is unique.
Note that $R$ need not be a domain. In case $R$ is a field, we can remove the assumption that $f(X)$ is monic. In the general case this assumption could be relaxed to “the leading coefficient of $f(X)$ is invertible”, but nothing really different would be obtained.
A: Use the fact that $\mathbb F_7[X]$ is euclidean. As a consequence, any polynomial $P \in \mathbb F_7[X]$ can be written as $P = (X^3 +2)Q + R$ for some $R$ such that $\deg R < 3$.
In particular, there are a,b,c in $\mathbb F_7$ such that $R = a + bX + cX^2$ and you can write $P \equiv a + bX + cX^2 \mod I$.
