Give an explicit ring isomorphism 
I want to give an explicit isomorphism between $\mathbb{F}_7[X]/(X^2+2X+2)$ and  $\mathbb{F}_7[X]/(X^2+X+3)$.

I think the way to do it would be to send a root $\alpha$ of $X^2+2X+2$ to the element $\beta$ of $\mathbb{F}_7[X]/(X^2+X+3)$ so that $\beta$ is a root of $(X^2+X+3)$. 

Hint: Note that $X^2+X+3=0$ and be rewritten as $4X^2+4X+12=0$, and then as $(2X+1)^2+4=0$.
Also, $X^2+2X+2=0$ can be rewritten as $(2X+2)^2+4=0$.  
