Finding isomorphisms between quotient rings of polynomials

Question

How do I find an isomorphism between the rings $\mathbb{F}_5[x] / (x^2 + x + 3)$ and $\mathbb{F}_5[x] / (x^2 + 3x + 2)$? The rings have 25 elements each so any exhaustive approach seems unlikely. We received the hint of using idempotents, but I don't know how to even find the idempotents of each ring without attempting to square all 50 elements across both rings.

Answer 1

Hint:

• $x^2+x+3\equiv_5(x-1)(x-3)$
• $x^2+3x+2\equiv_5(x+2)(x+1)$

Answer 2

Hint : an isomorphism has to send a root of $x^2+x+3$ to a root of $x^3+3x+2$. The roots of your first polynomial are $1,3$, and those of your second are $3,4$.