Results 1 to 6 of 6

Thread: quadratic again.. may be tricky

  1. #1
    Junior Member
    Joined
    Aug 2007
    Posts
    45

    quadratic again.. may be tricky

    hi, i think this one is a bit trickier.

    for an odd prime number "p", let $\displaystyle F_p$ 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 $\displaystyle F^n_p$ and why?

    any clues here please?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by joanne_q View Post
    So how many quadratic forms are there on the vector space $\displaystyle F^n_p$ and why?
    What do you mean by "how many quadradic forms..."? What do you mean by a "quadradic form"?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2007
    Posts
    45
    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Aug 2007
    Posts
    45
    here is the question as on the paper:

    for an odd prime number "p", let $\displaystyle F_p$ 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 $\displaystyle F^n_p$ and why?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Aug 2007
    Posts
    45
    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:
    ax + 2hxy + by = 1.

    i hope that makes it a bit more clearer..

    any ideas on how to solve the original question then?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Searching the internet reveals that a quadradic form is a symettric quadradic polynomial, for example, $\displaystyle x^2+y^2+4xy$. Thus we you are talking about a vector space $\displaystyle \mathbb{F}_p^n$ it seems to me are you trying to count all quadric forms of $\displaystyle x_1,...,x_n$ variables. So it has the form: $\displaystyle a_1x_1^2+...+a_nx_n^2+b_1x_1x_2+...$. Now just do a counting argument. Count all such coefficients. The $\displaystyle a_i$ for each $\displaystyle 1\leq i \leq n$ can be one of $\displaystyle 0,1,...,p-1$, thus in total we have $\displaystyle p^n$ combinations. Now the $\displaystyle b_i$ are all combinations of $\displaystyle x_i,x_j$ with $\displaystyle i\not = j$ there are $\displaystyle C_{n,2}=n(n-1)/2$ such possibilities. For each coefficient we have $\displaystyle p$ possibilities that we have $\displaystyle p^{n(n-1)/2}$. Altogether we have $\displaystyle p^{n(n-1)/2+n)}=p^{n(n+1)/2}$. But in counting this we have included that zero polynomial which is not a quadradic form so we just subtract $\displaystyle 1$ from that.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. some tricky eqn
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: Sep 28th 2009, 01:10 PM
  2. Replies: 10
    Last Post: May 6th 2009, 09:52 AM
  3. Replies: 1
    Last Post: Jun 12th 2008, 09:30 PM
  4. Replies: 5
    Last Post: May 30th 2008, 06:24 AM
  5. Ok then, here's a tricky one.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 12th 2008, 06:51 AM

Search Tags


/mathhelpforum @mathhelpforum