Hilbert symbol when K = the p-adic numbers

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.

Re: Hilbert symbol when K = the p-adic numbers

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Originally Posted by

**idontknowanything** 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.

The 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.

Re: Hilbert symbol when K = the p-adic numbers

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Originally Posted by

**Opalg**

I realize this, but every textbook I can get my hands on just asserts it without even half-attempting a proof, making me think it's still fairly simple.

Re: Hilbert symbol when K = the p-adic numbers

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.

Re: Hilbert symbol when K = the p-adic numbers

Can it be proved using the fact that $\displaystyle \left(\frac{a,b}{F}\right)\otimes\left(\frac{a,c}{ F}\right)=\left(\frac{a,bc}{F}\right)\otimes M_2(F)$ (where $\displaystyle \left(\frac{a,b}{F}\right)$ is a quaternion algebra over $\displaystyle F$)?

EDIT: This takes care of the cases where:

$\displaystyle (a,b) = (a,c) = 1$

$\displaystyle (a,b) \neq (a,c)$

since $\displaystyle (a,b)=1$ iff $\displaystyle \left(\frac{a,b}{F}\right)$ is split.

However, the case $\displaystyle (a,b) = (a,c) = -1$ would require proving that the tensor product of two non-trivial quaternion algebras in $\displaystyle \mathrm{Br}(\mathbb{Q}_p)$ is a matrix algebra, which seems non-trivial.