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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

Mathew
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1 Answers1

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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

mathma
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