The Arithmetic of Quaternion Algebra

3.4. NORM THEOREM AND STRONG APPROXIMATION THEOREM 61
A rapid method for obtaining the examples is to use the likeness in order to
avoid the study of {a, b}2, with remarking that if p is a prime number with p ≡
−1(mod4), then {−1, −p} has the discriminant p, finally for p ≡ 5mod(8), then
{−2, −p} has discriminant p. A little attempt allows to find easily a quaternion
algebra with a given discriminant, that is to say, two integer numbers , of which
the local Hilbert symbols are given in advance. For example,
{−1, 3}, d = 6; {3, 5}, d = 15; {−1, 7}, d = 14.
If p is a prime, p ≡ −1(mod4), then {2, p} has the discriminant 2p; if p ≡
5(mod8) , then {−2, p} has the discriminant 2p.
3.4 Norm theorem and strong approximation theorem
. The norm theorem was proved in 1936-1937. Hasse and Schilling [1], Schilling
[1],Maass [1], Eichler [3], [4] made contribution to its proof.
Its application to the euclidean order, and to the functional equation of L function
was made by Eichler [5]. The strong approximation theorem for for the
unit group with reduced norm 1 of the central simple algebra over number fields
is due to Kneser [1], [2], [3]. A recent article have proved this theorem in the
case of function field(Prasad [1]).
Theorem 3.4.1. (Norm theorem) Let KH be the set of the elements of K which
are positive for the infinite real place of K and ramified in H. Then KH = n(H).
Proof. The condition is natural since n(H) = R+. Conversely let x ∈ K ×
H ; we
construct a separable quadratic extension L/K such that :
– x ∈ n(L),
– for every place v ∈ Ram(H), Lv/Kv is a quadratic extension. Therefore, L
is isomorphic to a commutative subgroup of H by 3.8, and x ∈ n(H). This is
an exercise of using the approximation theorem and the lemma on polynomials.
Let S be a finite set of places of K. For the finite v we see that Hvcontains an
element of reduced norm πv. Since H is dense in Hv, we see that H contains
an element of reduced norm a uniform parameter of Kv, and by multiplying x
with n(h) for a suitable element h ∈ H, we can assume that for a finite set S of
places of K:
x is a unit for p ∈ S ∩ P .
We choose for every v ∈ S an extension Lv such that : – Lv − C if v is real,
– Lv is the unramified quadratic extension of Kv if v ∈ P ∩ S.
for every v ∈ S,it exists yv ∈ Lv of norm x. The minimal polynomial of yv over
Kv can be written as
pv(X) = X 2 − avX + x.
We choose a ∈ K very near to av if v ∈ S (and the same if one needs an integer
for every place of K excluding possibly one place v /∈ S), so that the polynomial
p(X) = X 2 + aX + x
is irreducible and defines an extension K � K(y) � K[X]/(p(X)) ⊂ Xs, such
that K(y)v = Lv, if v ∈ S.