The Arithmetic of Quaternion Algebra

2.1. CLASSIFICATION 25
Theorem 2.1.3. Let K be a non archimedean local field. then H = {Lnr, π} is
the unique quaternion field over K up to isomorphism. A finite extension F/K
neutralize H if and only if its degree [F : K] is even.
The second part of the theorem is an easy consequence of the first part. It
has two variants:
(1) H possess a F -representation if and only if [F : K] is even.
(2) ε(HF ) = ε(H) [F :K] .
The proof of the theorem is divided into several steps. Consider a quaternion
field H/K. We extend a valuation v of K to a valuation w of H. it shows Lnr
can be embedded in H. Using I.Corollary 2.2 and 2.4 we obtain H � {Lnr, π}.
The existence of the valuation w in addition gives the uniqueness of the maximal
order and the group structure of the normal ideals. We are going now to proceed
along this line. Reference: Serre [1].
Definition 2.3. A discrete valuation on a field X 2 is a mapping v : X × → Z
satisfying
(1) v(xy) = v(x) + v(y),
(2) v(x + y) ≥ inf(v(x), v(y)), with the equality if v(x) �= v(y) for every
x, y ∈ X × . An element u which has a positive minimal valuation is called a
uniform parameter of X. v can be extended to a mapping of X to Z ∪ ∞ by
setting v(0) = ∞. The set A = {x ∈ X|v(x) ≥ 0}is the discrete valuation ring
associated with v. Its unique prime ideal is M = Au = {x ∈ X|v(x) > 0}. The
field A/M is the residue field and the group A × = {x ∈ X|v(x) = 0} is the
unit group of A.
We choose a discrete valuation v of K; it can be suppose that v(K × ) = Z.
We define a mapping w : H × → Z by setting
w(h) = v ◦ n(h), (2.1)
where h ∈ H × and n : H × → K × is the reduced norm. The multiplicative
of the reduced norm (I,Lemma 1.1) implies w satisfies (1). We utilize a well
known fact that the local commutative field being the restriction of w to L
is a valuation if L/K is an extension of K contained in H. It follows then
w(h + k) − w(k) = w(hk −1 + 1) ≥ inf(w(hk −1 ), w(1) with the equality if
w(hk −1 ) �= w(1). From this we deduce that w satisfies (2). We have proved :
Lemma 2.1.4. The mapping w is a discrete valuation of H.
We denote the ring of the valuation W by O. For every finite extension
L/K contained in H, the intersection O ∩ L is the ring of the valuation of the
restriction of w to L. Therefore , O ∩ L is the integral ring RL of L. It follows
that O is an order consisting of all the integers of of H. We then have
Lemma 2.1.5. The ring O of valuation w is the unique maximal order of H.
Therefore, we deduce that every normal ideal of H is Two-sided.If u ∈ O is
a uniform parameter, P = Ou is the unique prime ideal of O. Thus the normal
ideals are of the form P n , n ∈ Z.
Lemma 2.1.6. The quadratic unramified extension Lnr/K of K is isometric
to a commutative subfield of H.
2 not necessary to be a commutative field