I'm working on a fairly simple problem about a field, but I want to know if the operations can be explicitly described.
Suppose $c$ is not a quadratic residue modulo $p$, and consider the quotient ring $\mathbb{F}_p[X]/(x^2-c)$. Now $x^2-c$ is irreducible over $\mathbb{F}_p$, so it generates a maximal ideal, and thus $\mathbb{F}_p[X]/(x^2-c)$ can be viewed as a 2-dimensional vector space over $\mathbb{F}_p$, and thus has order $p^2$. If I take $a$ to be a root of $x^2-c$ in some extension field, then I can view the elements of the field as $0,a,c,ac,c^2,ac^2,c^3,\dots,ac^{(p^2-3)/2},c^{(p^2-1)/2}$, for a total of $p^2$ elements.
However, I don't know how to actually state what addition and multiplication look like in this field. Is there a clever way to describe the operations explicitly? Thanks.