Isomorphism of rings (with parameters) Let $p$ be a prime number. For each choice of $a,b\in \mathbb{F}_p,$ let $F(a, b)$ be the ring $\mathbb{F}_p[X]/(X^2+aX+b).$ Find all possible choices of $(a, b), (a', b')\in \mathbb F_p \times \mathbb F_p$ such that $F(a, b) \cong F(a', b').$

HINT: Whatever $a$ and $b$, the ring will be a $\Bbb F_p$ vector-space of dimension $2$. As ring we have the following three possibilities for the quotient:


*

*A ring with a unique maximal ideal $\frak m$ such that $\frak m^2=(0)$.

*The ring $\Bbb F_p\times\Bbb F_p$.

*A field with $p^2$ elements.
These situations should be linked to the behaviour of the roots of the polynomial $X^2+aX+b$ to complete the answer.
Note that there's only one finite field of given cardinality up to isomorphism.
