The Arithmetic of Quaternion Algebra

24 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD
otherwise. We call ε(H) the Hasse invariant of H.
A variant of the above theorem is :
Quat(K) � {∓1}, if K �= C, Quat(C) � {1}.
Definition 2.2. If the characteristic of K is different from 2, and if a, b ∈ K × ,
the Hasse invariant of a, b is defined by
ε(a, b) = ε({a, b}),
where H = {a, b} is the quaternion algebra described in Ch.I.(3). The Hilbert symbol
of a, b is defined by
�
1, if ax
(a.b) =
2 + by2 − z2 = 0 has a non-trivial solution in K3 ,
, −1, otherwise
where the ”non-trivial solution” means a solution (x, y, z) �= (0, 0, 0).
A variant of the above theorem in characteristic different from 2 is expressed
as an equality between the Hilbert symbol and the Hasse invariant, and a variety
of properties of Hilbert symbol deduced from it.
Corollary 2.1.2. (Properties of Hilbert symbol). Let K is a local field of char.
different from 2.Let a, b, c, x, y ∈ K × . The Hilbert symbol {a, b} equals the Hasse
invariant ε(a, b). It satisfies the following properties:
(1) (ax 2 , by 2 ) = (a, b) (modulo the square),
(2) (a, b)(a, c) = (a, bc) (bilinearity),
(3) (a, b) = (b, a)(symmetry),
(4)(a,1-a) = 1 (symbol),
(5) (a, b) = 1 for all b ∈ K × implies a ∈ K ×2 (non degenerate),
(6)(a, b) = 1 is equivalent to one of the following properties:
−a ∈ n(K( √ b))
or b ∈ n(K( √ a)),
or −ax 2 + by 2 represents 1
Proof. The equation ax 2 + by 2 − z 2 = 0 has a non trivial solution in H 3 if
and only if the quadratic vector space V0 associated with pure quaternion of
{a, b} is isotropic. From ch.I, corollary 3.2, the space V0 is isotropic if and only
if {a, b}is isomorphic to a matrix algebra. Therefore (a, b) = 1 if and only if
ε(a, b) = 1. it follows (a, b) = ε(a, b). The properties (1),(2),(3),(4),(5),(6) are
the consequence of the earlier results.
(1),(3): Define the elements i, j by the formula I.1.(3) and replace i, j by xi, yj,
then by j, i.
(2). Use the tensor product(I,Thm 2.9).
(4),(6). Use the characterization of the matrix algebra (I,Corollary 2.4) and the
geometric consideration (I,Corollary 3.2)
(5).Obtain from that, all the quadratic extension of K can be included in the
quaternion field over K, if K �= C. This property will be proved more precisely
later(II,Corollary 1.9).
We suppose afterwards that K �= R, C. The theorem of classification has the
following very precise statement.