Wikipedia page on the Hilbert symbol states that the proof of bimultiplicativity is not straight forward, and requires the use of local class field theory.
How can I show that the Hilbert Symbol is bimiltuplicative, when the local field is the p-adic numbers? Everything I can find just sort of asserts bimultiplicativity without much proof, so I'm guessing it's pretty straight forward, but I haven't done much work with the p-adics so I'm a little unclear.
Moreover, for what primes p is it the case that there exists an element z of the p-adics such that (-1, z) = -1. That is, the the Hilbert symbol acts on -1 and z and evaluates to -1.
Serre's A Course in Arithmetic, pp. 19 - 21, contains a proof of the bilinearity of the Hilbert symbol when the local field is the p-adic numbers. The basic idea is to express the Hilbert symbol in terms of Legendre symbols and then use the multiplicative properties of the latter symbols.
I think that the cited Wikipedia article is, shall we say, misleading. It gives a definition of the Hilbert symbol for any local field, but that definition is given in most sources only for the p-adic numbers. The definition of the Hilbert symbol for an arbitrary local field is more complex (see, for example, Milne's notes on Class Field Theory: http://www.jmilne.org/math/CourseNotes/CFT310.pdf, pp. 88 and following). The proof of bilinearity for this more general definition is what apparently requires class field theory.
Can it be proved using the fact that (where is a quaternion algebra over )?
EDIT: This takes care of the cases where:
since iff is split.
However, the case would require proving that the tensor product of two non-trivial quaternion algebras in is a matrix algebra, which seems non-trivial.