Hilbert Symbol - Properties

Properties

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.