abstract algebra - Finding an isomorphism between these two finite fields

Let $\alpha$ be a root of $x^3 + x + 1 \in \mathbb{Z}_2[x]$ and $\beta$ a root of $x^3 + x^2 + 1 \in \mathbb{Z}_2[x]$. Then we know that

$$\mathbb{Z}_2(\alpha) \simeq \frac{\mathbb{Z}_2[x]}{(x^3 + x + 1)} \simeq \mathbb{F}_8 \simeq \frac{\mathbb{Z}_2[x]}{(x^3 + x^2 + 1)} \simeq \mathbb{Z}_2(\beta).$$

I need to find an explicit isomorphism $\mathbb{Z}_2(\alpha) \to \mathbb{Z}_2(\beta).$

I was thinking of finding a basis for $\mathbb{Z}_2(\alpha)$ and $\mathbb{Z}_2(\beta)$ over $\mathbb{Z}_2$.

I let $\left\{1, \alpha, \alpha^2\right\}$ and $\left\{1, \beta, \beta^2\right\}$ be these two bases. Now suppose I have a field morphism

$$\phi: \mathbb{Z}_2(\alpha) \to \mathbb{Z}_2(\beta)$$ which maps $1$ to $1$. how can I show that the image of $\alpha$, i.e. $\phi(\alpha)$, completely determines this map?