Quadratic forms with p-adic coefficient

[antonio85]

How can I found out in which p-adic fields a quadratic form represent 0?

For example in which p-adic fields does the form 3x²+7y²-15z² represent zero?

 

[MathematicalPhysicist]

You need to calculate Hasse Invariant and the discriminant of this form.

After that there's a theorem that gives conditions for which this form represents zero.
I can't recall the condition but I think Hasse Invariant, should be equal
(-1,-d(f)) where d(f) is the discriminant and (,) is Hilbert Symbol, after now after I checked my notebook, yes I am correct.

****, what a memory...