The Arithmetic of Quaternion Algebra

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4 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD it by H. It is the quaternion field defined over R with a = b = −1, called the quaternion field of Hamilton. It has a complex representation: � ′ z z H = −z ′ � , z, z z ′ ∈ C. The group consisting of the quaternion with reduced norm 1 is isomorphic to SU(2, C) and will be introduced for the geometric reason ( cf. the §3 geometry). Sometimes we call these quaternions the generalized quaternions ( compare with that of Hamilton) ,or hypercomplex numbers (perhaps it comes from the possible interpretation of the quaternions of Hamilton as a mixture of fields each being isomorphic to C ), but the general tendency is simply to say as quaternions. Exercises 1. zero divisor. Assume H/K a quaternion algebra over a commutative field K. An element x ∈ H is a zero divisor if and only if x �= 0 and there exists y ∈ H, y �= 0 such that xy = 0. Prove x is a zero divisor if and only if n(x) = 0. Prove if H contains at least one zero divisor, then H contains a zero divisor which is separable over K. 2. Multiplication of the quadratic forms. Prove that the product of two sums of 2 square integers is a sum of 2 squares integers. Prove the same result for the sums of 4 squares. Does the result still valid for the sums of 8 squares? Concerning with the last question, one can define the quasi-quaternions(Zelinsky [1]) or bi-quaternions(Benneton [3],[4]), or octonions of Cayley(Bourba Algebra, ch. 3,p.176) and study their arithmetic. 3. (Benneton [2]). Find the properties of the following matrix A, and from these to show a method to construct the matrices having the same properties: ⎛ 17 ⎜ 6 ⎝ 5 7 −14 −3 4 −1 −16 ⎞ 0 11 ⎟ 8 ⎠ 2 −10 9 13 . 4. prove an algebra of the matrices M(n, K) over a commutative field K is a K-central simple algebra 5. The mapping (h, k) → (t(hk) is a bilinear non-degenerate form on H (lemma 1.1). 6. underlinecharacteristic 2.If K is of char 2, then a quaternion algebra H/K is a central algebra of dimension 4 over K, such that there exists a couple (a, b) ∈ K × × K × and the elements i, j ∈ H satisfying i 2 + i = a, j 2 = b, ij = j(1 + i) such that H = K + Ki + Kj + Kij .
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.
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The Arithmetic of Quaternion Algebra

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