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How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$?

Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point $\infty_i$ to each class $C_i$, extend every line in $C_i$ to contain $\infty_i$, and finally connect all the "$\infty_i$" points into a line. I think I am on the correct track with this construction. Could someone use this start to perhaps complete the question?

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Looks like a good start to me (+1)! Isn't the rest case-by-case verification of the axioms of a projective plane? –  Jyrki Lahtonen Jul 18 '12 at 6:26
    
Yeah, but I am having trouble with the clarity of those cases. hence why I put the question here. :) –  Xuan Huang Jul 18 '12 at 7:03
    
Two lines in your projective plane are either both affine lines extended by a point or the other is the line at infinity. Treat those cases separately. The affine case splits further according to whether the two lines are parallel or not. –  Jyrki Lahtonen Jul 18 '12 at 7:22
    
I still can't put it together. How disappointing. –  Xuan Huang Jul 18 '12 at 15:38
    
Would you be able to show me these cases? –  Xuan Huang Jul 18 '12 at 21:53
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up vote 1 down vote accepted

As an example let us show that any two lines, $L_1$ and $L_2$, of your projective plane intersect at exactly one point.

If one of them, say $L_1$, consists of the points at infinity only, then $L_2$ intersects the affine plane along a line $L_2'$. Therefore $L_2$ contains exactly one infinite point, namely the one associated with all the affine lines parallel to $L_2'$. Therefore $L_1\cap L_2$ is a singleton.

If both $L_1$ and $L_2$ intersect the affine plane (along the lines $L_1'$ and $L_2'$), then there are two possibilities. If $L_1'$ and $L_2'$ are not parallel, then they intersect at a finite point. The same holds then for $L_1$ and $L_2$, as the infinite points are distinct (due to them not being parallel). OTOH, if $L_1'$ and $L_2$' are parallel, then they don't have any common affine points. But then the same point at infinity is tagged on them to get the lines $L_1$ and $L_2$, so the projective lines intersect at an infinite point.

The other axioms of a projective plane are verified in the same way. For example, you show that there is a line through any pair of points by splitting it into separate cases according to whether neither, one, or both of the points are infinite or affine.

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