What is the smallest affine plane not generated by a field? By "smallest" I mean has the least number of points. By "generated by a field", I mean planes of this form:
$X = \mathbb{F}^2$ (the set of points)
$$
L = \{\{(x,ax+b) \mid x \in \mathbb{F} \} \mid a,b \in \mathbb{F}\} \cup \{ \{(x,y) \mid y \in \mathbb{F} \} \mid x \in \mathbb{F} \} \quad (\text{set of lines})
$$
If you could list the lines as sets of points that would be ideal, thanks
Also if there are multiple smallest such planes, one example is sufficient
If you construct an affine plane axiomatically then you can see that if it has enough symmetries then it should come from a skew-field. In particular if your affine plane satisfies Desargues Theorem then it should come from a skew-field and if it satisfies Pappus Theorem then it comes from a field.
By Wedderburn's Little theorem you get that if you have a finite affine plane that satisfies Desargues, then it should come from a field.
If you drop the Desargues Theorem assumption, then you can see that the affine plane comes from a ternary ring. I don't know much about ternary rings or the orders where they exist, but if you find a finite ternary ring which is not a field then you can find an finite affine plane that does not come from a field.
You can look at Artin's Geometric Algebra, Chapter II for the details.
EDIT: Here is a list of Non-Desarguesian planes, some of them are finite:
https://en.wikipedia.org/wiki/Non-Desarguesian_plane