hi, i think this one is a bit trickier.
for an odd prime number "p", let be the field with "p" elements. i.e. the integers {0,....p-1} with addition and multiplication defined modulo "p".
So how many quadratic forms are there on the vector space and why?
any clues here please?
a quadratic form by the way would be the quadratic equation corresponding to the symmetric matrix :
so for example, a 2x2 matrix would produce the follow quadratic form:
ax² + 2hxy + by² = 1.
i hope that makes it a bit more clearer..
any ideas on how to solve the original question then?
Searching the internet reveals that a quadradic form is a symettric quadradic polynomial, for example, . Thus we you are talking about a vector space it seems to me are you trying to count all quadric forms of variables. So it has the form: . Now just do a counting argument. Count all such coefficients. The for each can be one of , thus in total we have combinations. Now the are all combinations of with there are such possibilities. For each coefficient we have possibilities that we have . Altogether we have . But in counting this we have included that zero polynomial which is not a quadradic form so we just subtract from that.