smallest affine plane not generated by a field

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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:



$X = mathbbF^2$ (the set of points)

$$
L = (x,ax+b) mid x in mathbbF mid a,b in mathbbF cup (x,y) mid y in mathbbF mid x in mathbbF quad (textset of lines)
$$
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




affine-geometry




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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:



    $X = mathbbF^2$ (the set of points)

    $$
    L = (x,ax+b) mid x in mathbbF mid a,b in mathbbF cup (x,y) mid y in mathbbF mid x in mathbbF quad (textset of lines)
    $$
    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




    affine-geometry




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
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      down vote

      favorite











      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:



      $X = mathbbF^2$ (the set of points)

      $$
      L = (x,ax+b) mid x in mathbbF mid a,b in mathbbF cup (x,y) mid y in mathbbF mid x in mathbbF quad (textset of lines)
      $$
      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




      affine-geometry




      share|cite|improve this question













      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:



      $X = mathbbF^2$ (the set of points)

      $$
      L = (x,ax+b) mid x in mathbbF mid a,b in mathbbF cup (x,y) mid y in mathbbF mid x in mathbbF quad (textset of lines)
      $$
      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





      affine-geometry





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 25 at 23:34
























      asked Jul 25 at 22:15









      mathew

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