Quadratic again. may be tricky

[mathusers]

hi, i think this one is a bit trickier.

for an odd prime number "p", let [itex]F_p[/itex] 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 [itex]F^n_p[/itex] and why?

any clues here please?

 

[CompuChip]

Can you write down the general form of such a quadratic form?
E.g., for n = 1, 2, 3 it would be
[tex]a x^2; \quad a x^2 + b x y + c y^2; \quad a x^2 + b x y + c x z + d y^2 + e y z + f z^2[/tex] --
do you know of a general formula?

(By the way, you probably don't need the general formula, just the number of terms occurring in it. So in the above example, 1, 3 and 6 respectively).