Consider an affine space $\mathcal{A} = \operatorname{AG}(n,\mathbb{K})$, with $\mathbb{K}$ a field. Show that $\mathcal{A}$ contains $\operatorname{AG}(2,2)$ if and only if $\operatorname{char}(\mathbb{K})=2.$

I have no idea how to start with this. I still tried something:

$\Rightarrow$: suppose $\operatorname{AG}(2,2) \subseteq \mathcal{A}$. This means that $\mathcal{A}$ contains $4$ different points $A,B,C,D$ of which no $3$ are collinear and $AB \parallel CD$, $AC \parallel BD$, $AD \parallel BC$. Now I'm stuck. I believe I'll have to introduce a multiplicative identity $e$ and show that $e+e = 0$.

$\Leftarrow$: suppose $\operatorname{char}(\mathbb{K}) = 2$. This means that $|\mathbb{K}| = 2^h$, with $h\ge 1$ because a line has at least $2$ points. Somehow I'll have to prove that $h=2$.

Can someone help? Thanks.


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