Given a polynomial ring over a field $F[x]$, I can factor, for example, the ideal generated by an irreducible polynomial $ax^2 + bx + c$: $F[x]/\left<ax^2 + bx + c\right>$, and guarantee that this factor ring is also a field.
My question concerns the structure of this factor ring. For example, if I consider the factor ring $Z_p[x] / \left<ax^2 + bx + c\right>$ for some irreducible polynomial $ax^2 + bx + c$, I can guarantee, for example, that this field has $p^2$ elements.
I am unsure why this is the case. My understanding is that the coset representitives of this factor ring are possible remainders by division by $ax^2 + bx + c$. Is this the right idea, and how would I know that two different remainders aren't in the same coset? Thanks.
Best Answer
The quotient you get is a two-dimensional vector space over your field. Can you show that any such vector space must have $p^2$ elements?