Thread: quadratic again.. may be tricky

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?

Originally Posted by joanne_q

So how many quadratic forms are there on the vector space and why?

What do you mean by "how many quadradic forms..."? What do you mean by a "quadradic form"?

thats the problem i have myself.
i dont quite understand the question so im trying to see if any of you guys have an idea of what i am meant to be doing here

here is the question as on the paper:

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", how many quadratic forms are there on the vector space and why?

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.