I am reading on polynomials, fields and rings. The example I am looking at right now is below:
$$(X^2+X+4)*(2X+3) \in \mathbb{F_7}[X] / (X^3 + X+ 1)$$ $$[(X^2+X+4)*(2X+3)] = 5X^2+2X+3$$ The last part of the following statement about the size of the ring is not intuitive to me:
Every polynomial in $\mathbb{F_7}[X] / (X^3 + X+ 1)$ will have a degree at most $2$ which means that the size of the resulting ring will be $7^3$ elements.
As far as I understand the reason why every polynomial, $q(x)$, in that field will have a degree at most $2$ is because: $$ \deg(q(x))< \deg(p(x)) = \deg(X^3+X+1) = 3$$
but I cannot grasp why the resulting ring will contain $7^3$ elements.
more precisely: The elements of $\mathbb{F}_7[X]/(X^3+x+1)$ are equivalence classes of polynomials. An every such equivalence class has exactly one polynomial of degree $\le2$.
Now every polynomial of degree $\le 2$ can be written as $aX^2+bX+c$ with $a,b,c\in\mathbb{F}_7$. So there are are exactly $7$ possibilities for each of the coefficients $a$, $b$ and $c$. So there are $7\cdot7\cdot7=7^3$ possibilities in total.