The Arithmetic of Quaternion Algebra

1.2. THE THEOREM OF AUTOMORPHISMS AND THE NEUTRALIZING FIELDS5
1.2 The theorem of automorphisms and the neutralizing
fields
In this section it includes the applications of the fundamental theorems in the
central simple algebra to the quaternion algebra. these theorems can be found in
Bourbaki[2], Reiner [1], Blanchard [1], Deuring [1]. As in the next two chapters,
in this section we prefer to follow the book of Weil[1]. Let H/K be a quaternion
algebra.
Theorem 1.2.1. (Automorphisms, theorem of Skolem-Noether). Assume L, L ′
be two K-commutative algebra over K, contained in a quaternion algebra H/K.
Then every K-isomorphism of L to L ′ can be extended to an inner automorphism
of H. In particular, the K-automorphisms of H are inner automorphism.
Recall that, the inner automorphism of H is an automorphism k ↦→ hkh −1 , k ∈
H, associated with the invertible elements h ∈ H. Before proving this important
theorem, we are going to give a number of its applications.
Corollary 1.2.2. For every separable quadratic algebra L/K contained in H,
there exists θ ∈ K × such that H = {L, θ}.
. There is an element u ∈ H × which induces on L the nontrivial Kautomorphism
by the inner automorphism. We verify that t(u) = 0 (cf. §3),
then u 2 = θ ∈ K. We have also realized H in the form {L, θ}.
Corollary 1.2.3. . The group Aut(H) of the K-automorphisms of H is isomorphic
to the quotient group H × /K × . If L satisfy the corollary 2.2, the subgroup
Aut(H, L) consisting of the automorphisms fixing L globally is isomorphism to
(L × ∪ uL × )/K × , therefore the subgroup of the automorphisms fixing L is isomorphism
to L × /K × exactly.
.
Corollary 1.2.4. (Characterization of matrix algebra) A quaternion algebra
is either a field or isomorphic to a matrix algebra M(2, K). The quaternion
algebra {L, θ} is isomorphic to M(2, K) if and only if L is not a field or if
θ ∈ n(L).
Proof. .If L is not a field, it is clear that L, θ is isomorphic to M(2, K) (cf.
the passage in §1 concerning with the quaternion algebras over the separably
closed fields). We then suppose L is a field.We shall prove that, if H is not a
field, then θ ∈ n(L). We choose an element h = m1 + m2u with reduced norm
zero. We then have that 0 = n(m1) + θn(m2) and n(m1) = 0 is equivalent to
n(m2). Since L is a field, the property h �= 0 implies both m1, m2 are not zero,
therefore θ ∈ n(L). We shall show θ ∈ n(L) if and only if {L, θ} is isomorphic
to M(2, K). If θ ∈ n(L), it exists in H an element with its square being 1, but
different from ±1,and then a zero divisor. We choose in H a zero divisor which
is separable over K (see exercise 1.1), and denote it by x. Set L ′ = K(x). The
corollary show us that H = {L ′ , θ ′ }. Since L ′ is not a field, H is isomorphic
to M(2, K). If θ /∈ n(L), the non-zero elements of H have a non-zero reduced
norm and H is a field.