I have to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$.
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To show this do I have to take first an element of $\mathbb{F}_{2^2}$ and show that it belongs also in $\mathbb{Z}_2(a)$??
Let $x\in \mathbb{F}_{2^2}$.
$x^{2^2}=x\Rightarrow x^{4}=x \Rightarrow x^{3}=1$
But how could I continue??