The Arithmetic of Quaternion Algebra

Chapter 1
Quaternion Algebra over a
field
In this chapter K always denotes a commutative field of arbitrary characteristics
if no particular mention, and Ks is the separable closure of K.
1.1 Quaternion Algebra
Definition 1.1. A quaternion algebra H with center K is a central algebra over
K of dimension 4, such that there exists a separable algebra of dimension 2 over
K, and an invertible element θ in K with H = L + Lu, where u ∈ H satisfies
u 2 = θ, um = mu (1.1)
for all m ∈ L, and where m ↦→ m is the nontrivial K-automorphism of L.
We denote H sometimes by {L, θ}, but H does not determine this pair
{L, θ} uniquely. For example, it is clear that one could replace θ by θmm, if
m is an element of L such that mm �= 0. the element u is not determined
by (1). If m ∈ L is an element with mm = 1, we could replace u by mu.
the definition can be used in the case of arbitrary character. .we can verify
easily that H�K is a central simple algebra, i.e. an algebra with center K
and without any nontrivial two-sided ideal.conversely, we can prove that every
cental simple algebra of dimension 4 over K is a quaternion algebra.The rule of
multiplication is deduced from (1). If m ∈ L, we have
(m1 + m2u)(m3 + m4u) = (m1m3 + m2m4θ) + (m1m4 + m2m3)u. (1.2)
Definition 1.2. The conjugation is the K-endomorphism of H: h → h, which
is the extension of the nontrivial K-automorphism of L defined by u = −u.
It is easy to verify it is an involutive anti-automorphism of H. It can be
expressed by the following relations: if h, k ∈ H and a, b ∈ K, we have
ah + bk = ah + bk, h = h, hk = kh.
Definition 1.3. Assume h ∈ H. The reduced trace of h is t(h) = h + h. The
reduced norm of h is n(h) = hh.
1