Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings?

Question

Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings? If so, write down an explicit isomorphism. If not, prove they are not.

My Try:

Since $x^2+2$ is irreducible in $\mathbb{F}_5[x]$, and $y^2+y+1$ is irreducible in $\mathbb{F}_5[y]$, both $R_1$ and $R_2$ are fields. Moreover, $O(R_1)=25=O(R_2)$. Since for a given prime $p$ and integer $n$ there is a unique field with $p^n$ elements, $R_1$ and $R_2$ are isomorphic. But how can I write an explicit isomorphism? Can somebody please help me to find it?

Answer 1

Hint: $(y + 3)^2 + 2 = y^2 + y + 1$ as elements of $\mathbb F_5[y]$.

Answer 2

The start of your argument is fine. For the explicit isomorphism note that $x$ is a square root of $-2$ in the first, now find one in the second too.

To this end note that $y^2 + y +1 = (y^2+ y -1) +2 = (y^2+ 4y +4) +2 = (y+2)^2 +2 $.