Numbers of vectors in a vector space over a finite field, with different multiplication

Question

I had a recent question in an assignment that I couldn't complete. We are given the following:

• $q$ is an odd prime power.

• $(F,+,\cdot)=\text{GF}\left(q^2\right)$.

• $K$ is the $q$ element subfield of $F$.

• $\square=\text{non-zero squares of F}= \{ \alpha^{2i} \} $, where $\alpha$ is a primitive root of $\mathrm{GF}(q^2)$.

• $\boxtimes=\text{non-squares of F}= \{ \alpha^{2i+1} \} $

(i.e. $F=\square\sqcup\boxtimes\sqcup 0 $)

• $o \colon F \times F \rightarrow F$ is given by $$a o b = \left\{ \begin{array}{lr} ab & a \in \square \\ ab^q & a \in \boxtimes \\ 0 & a = 0 \end{array}\right.$$ So basically it's regular multiplication with the Frobenius automorphism (sometimes).

For previous parts of the question, I've determined:

1. $N$ = left near field $(F,+,o)$

2. $N$ is a vector space over $K$

Now I'm asked to find the number of points in a set $P$, where each point is defined as:

$P$ = set of triples (x,y,z), not all 0, identified up to scalar multiplication of N{0} i.e. $(x,y,z) == (n o x, n o y, n o z)$ $\forall n \in N \{ 0 \}$

So $P$ is the vector space $N^3$

I was hoping to use the follow: for a $n$ dimensional vector space over a finite field of order $q$, we have

$\left( \begin{array}{c} n \\ k \end{array} \right)_q$ k-dimensional subspaces.

(http://en.wikipedia.org/wiki/Gaussian_binomial_coefficient under Applications). But I'm unsure how to prove its allowed (or even if it is)

Answer 1

Yes. The number of elements of P agrees with the usual formula for the number of elements of projective space over a field. This follows by Orbit-Stabilizer, and the fact that the groups and stabilizers have the same size.

Let K be a finite field of size q, where q is an odd prime power, and let F be a finite field of size q² containing K. We define a new (unital, associative, but only partially distributive) multiplication on F, and call the resulting structure N. N is a 2-dimensional vector space over K and is the (regular) Dickson near-field construction.

In particular, for every nonzero a in N, the map μ_a:N→N:x↦(a o x) is a K-automorphism of the underlying vector space of N. The set of μ_a for nonzero a in N forms a group, and P is the set of orbits of the nonzero elements of N³ (a 6-dimensional K-vector space) under this action of μ.

The important feature of μ is that it is sharply transitive, that is, it acts regularly on the nonzero elements of N. If (x,y,z) is a triple of nonzero elements, then μ acts on each component regularly, and the orbit has size |N|−1. If one of the components is 0, say z=0, then in effect μ is only acting on (x,y), and again the orbit has size |N|−1. The only time the argument fails is if x=y=z=0, but this point is discarded for P anyways. In other words, every orbit of μ on N³−0 has size |N|−1, and so the number of orbits is (|N|³−1)/(|N|−1) = q⁴ + q² + 1, just like for the projective space over F.