Consider the polynomial ring $\Bbb F_m[x]$, and two polynomials $f,g $ in $\Bbb F_m[x]$.
Is there any necessary and sufficient conditions for $f,g$ such that $\Bbb F_m[x]/(f) \cong \Bbb F_m[x]/(g)$?
Or if there are two particular $f,g$ in $\Bbb F_m[x]$, how to check that the quotient rings generated by $f,g$ are isomorphic? I have this question when I encounter the following question:
Are $\Bbb F_3[x]/(x^3+x^2+x+1), \Bbb F_3[x]/(x^3-x^2+x-1) $ isomorphic?
I could not see a suitable isomorphism so I assume they are not isomorphic, yet cannot come to a contradiction so far.
Any help for the two problems is appreciated, thanks.