Find the cardinality of $\mathbb{F}_2$ adjoin a root of $X^4 + X + 1$

Question

Consider the irreducible polynomial $g = X^4 + X + 1$ over $\mathbb{F}_2$ and let $E$ be the extension of $\mathbb{F}_2 =\{0,1\}$ with a root $\alpha$ of $g$.

How many elements does $E$ have?

I do not know how to go about this question, please help.

Answer 1

We have $\mathbb{F}_2[\alpha]\cong \mathbb{F}_2[X]/(X^4+X+1)=\{ a_0+a_1 X+a_2 X^2+a_3 X^3 \mid X^4=X+1, a_i \in \{ 0, 1\} \}.$

Thus

$|\mathbb{F}_2[\alpha]|=|\{(a_0,a_1,a_2,a_3), a_i \in \{ 0, 1\} \}|=2^4.$