Mathematics 502
Homework (due Feb 14)
A. Hulpke
11) Show that in every projective plane there are four points, no three of which are collinear.
Note: One can show that a projective plane also can be characterized by the following set of axioms
which are invariant under duality:
1. Every two points are incident with a unique line.
2. Every two lines are incident with a unique point.
3. There are four points, no two collinear.
12)
Show that every projective plane of order 3 must be isomorphic to PG(2, 3).
13) The polynomial x2 + 1 is irreducible over the field with 3 elements. We can thus generate the
field with 9 elements via this polynomial. We denote the element represented by x by i as it is a
root of x2 + 1. We therefore get the following arithmetic tables:
+
0
1
−1
i
i+1
i−1
−i
−i + 1
−i − 1
0
1
−1
i i+1 i−1
−i −i + 1 −i − 1
0
1
−1
i i+1 i−1
−i −i + 1 −i − 1
1
−1
0 i+1 i−1
i −i + 1 −i − 1
−i
−1
0
1 i−1
i i + 1 −i − 1
−i −i + 1
i i+1 i−1
−i −i + 1 −i − 1
0
1
−1
i+1 i−1
i −i + 1 −i − 1
−i
1
−1
0
i−1
i i + 1 −i − 1
−i −i + 1
−1
0
1
−i −i + 1 −i − 1
0
1
−1
i i+1 i−1
−i + 1 −i − 1
−i
1
−1
0 i+1 i−1
i
−i − 1
−i −i + 1
−1
0
1 i−1
i i+1
·
0
1
−1
i
i+1
i−1
−i
−i + 1
−i − 1
0
1
2
i i+1 i+2
2∗i 2∗i+1 2∗i+2
0
0
0
0
0
0
0
0
0
0
1
−1
i i+1 i−1
−i −i + 1 −i − 1
0
−1
1
−i −i − 1 −i + 1
i
i−1
i+1
0
i
−i
−1 i − 1 −i − 1
1
i + 1 −i + 1
0 i + 1 −i − 1 i − 1
−i
1 −i + 1
−1
i
0 i − 1 −i + 1 −i − 1
1
i i+1
−i
−1
0
−i
i
1 −i + 1 i + 1
−1 −i − 1
i−1
0 −i + 1 i − 1 i + 1
−1
−i −i − 1
i
1
0 −i − 1 i + 1 −i + 1
i
−1 i − 1
1
−i
a) We define a “conjugation” ¯· by a + bi = a − bi. Show that for every x ∈ F we have that x = x3 .
(This is called a “Frobenius automorphism”.)
b) Show that x + y = x + y and x · y = x · y.
c) Determine the (four) nonzero squares in F.

a=0
 0
a · b b is a square . Show that ? is
d) We define a new multiplication ? on F by setting a ? b :=

a · b else
associative but not commutative.
e) Show that ? is right-distributive (i.e. (a + b) ? c = a ? c + b ? c) but not left-distributive.
f) Show that every element of F∗ := F \ {0} has a left inverse wrt. ?. (Thus — with extra work —
(F∗ , ?) is a group.)
Note: We have now established that (F, +, ?) is not a field, but “pretty close”. This kind of structure
is called a near-field.
g) Denote the elements of F by f 0 , . . . , f 8 with f 0 = 0. We define 8 matrices A1 , . . . , A8 by setting
(Am )i, j = f m ? f i + f j with 0 ≤ i, j ≤ 8 (analogous to proposition (6.6.1) in the book). Show that this
defines a set of 8 MOLS (and thus a projective plane we shall call Ω).
14)
Show that any set of m − 2 MOLS of order m can be enlarged to a set of m − 1 MOLS.
15∗ ) Prove Desargues theorem (9.5.3) for PG(2, q). (“Concurrent” means that the lines a1 a2 b1 b2
and c1 c2 intersect in the same point.)