How can I show that the Hilbert Symbol is bimultiplicative, 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 $-1$.