Polynomial Ring Divided By Principal Ideal

Question

Let $F_5[X]$ be the polynomial ring over $F_5$ and $I = <X^2+X+1> $. Show that any element of $\frac{F_5[X]}{I}$ can be written as $a+bX+I$ where $a,b$ are in $F_5$.

I guess I can be written as $f(x)(X^2+X+1)$ but I am unsure how to divide?

Thanks for your help!

Answer 1

Hint: In $F_5[X]/I$ you have $X^2=4X+4$.

Answer 2

Let $P\in F_5[X]$ divides $P$ by $X^2+X+1$ you have $P=U(X^2+X+1)+a+bX$. Then the image of $P$ and $a+bX$ in $F_5[X]/(X^2+X+1)$ are equal.