How to determine the cardinal of $\mathbb{F}_p[X] / (X^2 + X + 1)$

Question

Let $p$ be a prime number and let's consider the quotient ring $$A=\mathbb{F}_p[X] / (X^2 + X + 1) $$ What is the cardinal of $A$?

I am unable to solve this. Su I could state that $\text{Card(A)}\leq p$. Now, if I would be able to find a homomorphism between $\mathbb{F}_p[X]$ and some other ring $B$ such that $\text{Ker(f)}= (X^2 + X + 1) $, I would be able to state that $\mathbb{F}_p[X] / (X^2 + X + 1) = \mathbb{F}_p[X] /\text{Ker(f)}$ is isomorphic to $\text{Im}(f)$ and deduce the cardinal from this fact.

I could maybe use group actions, but not sure how.

Can anyone help me?

Answer 1

Elements of A would be of type $aX+b$ where $a,b \in \mathbb{F}_p$ for which we have $p^2$ numbers of choices for $a$ and $b$.

Answer 2

HINT:
Any element of this quotient group is of the form $(ax+b)+[x^2+x+1].$
Now check possible values for $a$ and $b.$

Answer 3

There is a basis consisting of polynomials with degree less than $2$: $\{1,x\}$ in particular consists of a basis for the vector space over $\mathbb F_p$. In particular, each of these guys is isomorphic to $\mathbb F_p$, so $|\mathbb F_p|^2$ is the cardinality of the quotient ring

Answer 4

In general, for a field $k$ and a polynomial $f$ of degree $d$ over $k$, $$R=k[X]/(f(X))$$ is a $d$-dimensional vector space over $k$ (spanned by the images of $1$, $X,\ldots,X^{d-1}$). In particular if $k$ is finite of order $q$, then $|R|=q^d$.