Irreducible Polynomials: How many elements are in E?

Question

Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$ and let $E$ be the extension of $F_2 =$ {0, 1} with a root $α$ of $g$.

How many elements does $E$ have?

Im really not sure how to go by this question so any help will be appreciated

Answer 1

Hint:

$E$ is an $\mathbf F_2$-vector space, isomorphic to the quotient ring $\mathbf F_2 (f]/(g)$. What is its dimension?

Answer 2

We have $\mathbb Z_2[x]/(g(x))\cong \mathbb Z_2(\alpha); \alpha=x+(g(x))$ and $\{1,\alpha,\alpha^2,\alpha^3 \} $ is a basis of $\mathbb Z_2(\alpha)$ over $\mathbb Z_2$, i.e. $\mathbb Z_2(\alpha)=\{0,1,\alpha,\alpha^2,\alpha^3,1+\alpha,1+\alpha^2,1+\alpha^3,\alpha+\alpha^2,\alpha+\alpha^3, \alpha^2+\alpha^3,1+\alpha+\alpha^2,1+\alpha^2+\alpha^3,1+\alpha+\alpha^3,\alpha+\alpha^2+\alpha^3,1+\alpha+\alpha^2+\alpha^3 \}$