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Math Help - Hilbert symbol when K = the p-adic numbers

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

    Quote Originally Posted by idontknowanything View Post
    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.
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    Re: Hilbert symbol when K = the p-adic numbers

    Quote Originally Posted by Opalg View Post
    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.
    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.
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    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.
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    Re: Hilbert symbol when K = the p-adic numbers

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

    EDIT: This takes care of the cases where:

    (a,b) = (a,c) = 1
    (a,b) \neq (a,c)

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

    However, the case (a,b) = (a,c) = -1 would require proving that the tensor product of two non-trivial quaternion algebras in \mathrm{Br}(\mathbb{Q}_p) is a matrix algebra, which seems non-trivial.
    Last edited by redsoxfan325; December 6th 2011 at 10:34 PM.
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