$\overline{\mathbb{Z_2}}$ closure of $\mathbb{Z_2}$, $\alpha, \beta$ roots of $x^3+x^2+1$, $x^3+x+1$. Show $\mathbb{Z}(\alpha) = \mathbb{Z}(\beta)$.

Question

Let $\overline{\mathbb{Z_2}}$ be an algebraic closure of $\mathbb{Z_2}$, and let $\alpha, \beta \in \overline{\mathbb{Z_2}}$ be zeroes of $x^3+x^2+1$ and of $x^3+x+1$, respectively. Show that $\mathbb{Z}(\alpha) = \mathbb{Z}(\beta)$.

The solution:

Theorem 33.3:

First of all, shouldn't $\mathbb{Z_2}$ have $2$ elements? Why $\mathbb{Z}(\alpha)$ and $\mathbb{Z}(\beta)$ are extensions of degree $3$?

Answer 1

If $K$ is any field over which $x^3+x^2+1$ is irreducible, then $K(\alpha)\cong K(\beta)$ where $\alpha$ and $\beta$ are zeroes of $x^3+x^2+1$ and $x^3+x+1$. That is because $1/\alpha$ is a zero of $x^3+x+1$.