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Math Help - Isotropic forms in field of p-adic numbers

  1. #1
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    Isotropic forms in field of p-adic numbers

    Hi,

    for which primes p are the quadratic forms

    7x^2-y^2-5z^2

    and

    w^2+x^2+3y^2+11z^2

    isotropic over the field over p-adic numbers \mathbb Q_p ?

    A abbreviated form for the two quadratic forms above is \left \langle 7,-1,5 \right \rangle and \left \langle 1,1,3,11 \right \rangle.

    Define \varepsilon (\left \langle a_1,...,a_n \right \rangle):=\prod_{1\leq i< j\leq n} (a_i,a_j) \in \left \{ \pm 1 \right \}.

    I can use the fact that a regular quadratic form \phi over a field F is isotropic if dim(\phi)=3 and
    \varepsilon(\phi)=(-1,-d).

    [ (. , .) denotes the Hilbert symbol and d:=disc(\phi) the discriminant of \phi.]

    Moreover, a regular quadratic form \phi over a field F is isotropic if dim(\phi)=4 and
    d \neq 1 or { d=1 and \varepsilon(\phi)=(-1,-1)}.

    d \neq 1 means actually that the residue classes of d and -1 are not equal modulo F^*^2.

    Let's come to the first form:
    \varepsilon (\left \langle 7,-1,-5 \right \rangle)=(7,-1)(7,-5)(-1,-5)

    d=disc(\left \langle 7,-1,-5 \right \rangle)=35 , therefore -d=-35.

    Now I have to check for which primes p
    (7,-1)(7,-5)(-1,-5) = (-1,-35) in \mathbb Q_p.

    But I don't know how to solve it.

    Please can you help me?

    Thanks!

    Regards

    Alexander
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  2. #2
    Newbie
    Joined
    Jan 2010
    Posts
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    Re: Isotropic forms in field of p-adic numbers

    Hello,

    this problem is still unsolved.
    If you have any hints or ideas, it would be great if you would answer.

    Bye,
    Alexander
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