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Concept
Hilbert symbol
Summary
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
The Hilbert symbol has been generalized to higher local fields.
Over a local field K whose multiplicative group of non-zero elements is K×,
the quadratic Hilbert symbol is the function (–, –) from K× × K× to {−1,1} defined by
Equivalently, if and only if is equal to the norm of an element of the quadratic extension page 110.
The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
If a is a square, then (a, b) = 1 for all b.
For all a,b in K×, (a, b) = (b, a).
For any a in K× such that a−1 is also in K×, we have (a, 1−a) = 1.
The (bi)multiplicativity, i.e.,
(a, b1b2) = (a, b1)·(a, b2)
for any a, b1 and b2 in K× is, however, more difficult to prove, and requires the development of local class field theory.
The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group , which is by definition
K× ⊗ K× / (a ⊗ (1−a), a ∈ K× \ {1})
By the first property it even factors over . This is the first step towards the Milnor conjecture.
The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules , , . In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
For a place v of the rational number field and rational numbers a, b we let (a, b)v denote the value of the Hilbert symbol in the corresponding completion Qv.
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