Mathematics 502
Homework (due Feb 7)
A. Hulpke
5) Let Fq (n) be the total number of subspaces of and n-dimensional vector space over the
field with q elements. Prove that Fq (0) = 1, Fq (1) = 2 and
Fq (n + 1) = 2Fq (n) + (qn − 1)Fq (n − 1)
for n ≥ 1. (Hint: use (9.2.3) and (9.2.4) from the book.)
6) Show that for every prime power q there is a number 0 < c(q) < 1 such that the
probability of an n × n matrix over GF(q) being invertible tends to c(q) as n → ∞.
7∗ ) The polynomials x3 + x + 1 and x3 + x2 + 1 are both irreducible over GF(2). Show
explicitly that the fields of order 8 constructed via these two polynomials are isomorphic.
8) Let π be a projective plane. We form a new geometry ρ by setting the points of ρ to
be the lines of π and lines of ρ being the points of π . A point L lies on a line a in ρ iff a lies
on L in π . We call ρ the dual of π .
a) Show that ρ is a projective plane.
b) Show that PG(2,2) is isomorphic (i.e. there is a bijection of points preserving collinearity) to its dual.
c) Show that in a projective plane every line contains at least 3 points.
9)
Construct (i.e. describe the lines) a projective plane of order 3.
10) Quaternions were defined by H AMILTON in an attempt to generalize complex numbers. We take three objects i, j, k which fulfill the rules:
i2 = j2 = k2 = i jk = −1
i j = k, ji = −k
jk = i, k j = −i
ki = j, ik = − j
The quaternions are all linear combinations q = a + bi + c j + dk with a, b, c, d ∈ R. Describe
a formula for multiplication in the quaternions.
Problems marked with ∗ are bonus problems for extra credit.