Express elements of field in terms of residue class

Question

The field $K$ with $9$ elements is constructed as the quotient $F_3 [X]/(X^2 + 1)$. Let a be the class of $X$ $($so $a = X + (X^2 + 1))$. Here $F_3$ is short for $Z/3Z$.

Express all the elements of $K$ in terms of $a$.

Answer 1

Hint:

For any field $F$ any class modulo $P(X)$in $F[X]$ is represented uniquely by its remainder after division by $P(X)$.