Good day, I found a question which finds isomorphisms between 2 fields
Find an explicit isomorphism of rings
I am, however, interested in finding out how to construct such an isomorphism if the domain and codomain weren't fields. Notably,
$\lambda:\mathbb{F}_5[x]/(x^2+x+3)\rightarrow\mathbb{F}_5[x]/(x^2+3x+2)$
In lectures, we were told a little about this and were hinted to use idempotents but I am not quite clear on how exactly I am supposed to use them. Can I apply a similar technique as in the question that I linked? Except I cant seem to make sense of how the author of the answer got the relations for $a$ and $b$.
What I found, is that using this rather universal method, such problems can be solved for fields and non-fields
Define a function $\lambda:\mathbb{F}_5[x]/(x^2+x+3)\rightarrow\mathbb{F}_5[x]/(x^2+3x+2)$ s.t.
$\lambda (x)=ax+b$
$\lambda(0)=0$
$\lambda(a)=a$
We can send the polynomial in the codomain to $\equiv0$ $mod(\text{polynomial in codmain})$ (notation is probably wrong but the idea will be clear below)
$\lambda(x^2+x+3)\equiv 0 \ mod(x^2+3x+2)$. Which means
$$(ax+b)^2+(ax+b)+3 = a^2(x^2+3x+2)$$
Should get something like $$2ab+a=3a^2 \text{ and } b^2+b+3=2a^2$$
Just plug in elements of $\Bbb F_5$ to see which ones solve the equations. I got, say, $a=3, b=4$
So, a possible isomorphism is $$\lambda(x)= 3x+4$$ All that is left is to show injectivity, surjectivity and the fact that this is a homomorphism.
It should all check out.