The Arithmetic of Quaternion Algebra

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8 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD b)Every element τ in the group Gal(Ks/K) of the K-automorphism of Ks induces a permutation r of 1, ..., n: τ.σi = σ r(i), it is a K-isomorphism of Hi on H r(i) , by the restriction of the mapping : τ(h ⊗ k) = h ⊗ τ(k), h ∈ H, k ∈ Ks, and a K-isomorphism of D. c) The elements of D being invariant by Gal(Ks/K) constitute a central simple algebra of dimension 4 n over K. The construction below proceeds naturally when H is a central simple L-algebra. The algebra constructed over K is denoted by Cor L/K(H). It corresponds to the co-restriction mapping under the cohomology interpretation of the Brauer groups. 2. Let L/K be a separable extension of K of degree 2, and m → m be the nontrivial K-automorphism � � of L. Prove m n a) The set {g = } forms a K-algebra which is isomorphic to n m M(2, K). b) If g is invertible, prove that g−1 � � is conjugate to g by an element of the r 0 form with r ∈ L 0 r ⋆ . 1.3 Geometry In this section we assume the characteristic of the field is different from 2. For every quaternion algebra H/K we use H0 to denote the set of the quaternions with zero reduced trace. The reduced norm provides the K-vector space V, V0, the subjacent spaces of H, H0 respectively, a non-degenerate quadratic form. We denote the associated bilinear form by < h, k > for h, k ∈ V or v0. It is defined by < h, k >= t(hk, from it we deduce < h, h >= 2n(h). If the elements h, k belong to V0 we have a simple formula < h, k >= −(hk + kh). We see also that the product of two elements of H0 is an element of H0 if and only if these elements anti-commute (hk = −kh), It is also equivalent to that, these two elements are orthogonal in H0. We now study the quaternion algebra with the point of view of their quadratic spaces. Lemma 1.3.1. Let H, H ′ be two quaternion algebras over K, and V, V0, V ′ V ′ 0 be the correspondent quadratic spaces respectively. The following properties are equivalent: (1) H and H ′ are isomorphic, (2) V and V ′ are isomorphic. (3) V0 and V ′ 0 are isomorphism. Proof. (1) implies (2), because an automorphism preserving the norm induces an isomorphism. (2) implies (3) by the theorem of Witt, and the the orthogonal decomposition V = K + V0, which is deduced from (3) in §1. (3) implies (1), because an isometry preserves the orthogonality, hence if i, j ∈ H satisfy (3) in §3, then f(i) and f(j) satisfy the same relations and H = H ′ .
1.3. GEOMETRY 9 Corollary 1.3.2. The following properties are equivalent: (1) H is isomorphic to M(2, K), (2) V is an isotropic quadratic space, (3) V0 is an isotropic space, (4) the quadratic form ax 2 + by 2 represents 1. Proof. (1) is equivalent to (2) comes from the characterization of matrix algebra considered in §1. (1) is equivalent to (3), it is clear too. (4) implies (1), since the element ix + jy is of square 1 if ax 2 + by 2 = 1, and it is different from ∓1 , then His not a field.(3) implies (4) since if ax 2 + by 2 − abz 2 = 0 with z �= 0,it is clear that ax 2 + by 2 represent 1, and otherwise b ∈ −aK 2 , and the quadratic form ax 2 + by 2 is equivalent to a(x 2 − by 2 ) which represents 1. According to the theorem of Cartan (Dieudonné[1]), every isometries of a K vector space of finite dimension m equipped with a quadratic form is the product of at most m symmetries. The theorem shows that the proper isometries(i.e. of determinant 1) of V0 are the products of two symmetries of V0. The symmetry of V of a non isotropic vector q can be written as h → sq(h) = h − qt(hq)/n(q) = −qhq −1 , h ∈ H . If q, h are in V0, this symmetry is simply defined by sq(h) = −qhq −1 . The product of two symmetries sq, sr of V0 is defined by sqsr(h) = qrh(qr) −1 . Conversely, we shall prove every inner automorphism of H induces on V0 a proper isometry. If the isometry induces on V0 by an inner automorphism is not proper, then there would exist r ∈ H × such that for x ∈ V0 the image of x equals −rxr −1 . We then deduce from this that h → rhr −1 is an inner automorphism, it is absurd. We have proved the following theorem too. Theorem 1.3.3. The proper isometries of V0 are obtained by the restriction of the inner automorphism of H to the quaternions with zero trace. The group of proper isometries of V0 is isomorphic to H × /K × . The last assertion comes from corollary 2.3. By the same token we demonstrated a quaternion can be written as the product of two quaternions exactly by an element of K. The theorem allows us to rediscover some classical isomorphisms between the orthogonal groups and some quaternion groups. We denote the group GL(2, K)/K × by P GL(2, K); the proper isometric group of the quadratic form x 2 − y 2 − z 2 over K by SO(1, 2, K) ; the rotation group SO(3, R) of R � has a non-trivial covering of degree 2, denoted by Spin(3, R). If H/K is a quaternion algebra, H 1 denotes the kernel of the reduced norm. Theorem 1.3.4. We have the isomorphisms: 1) P GL(2, K) � SO(1, 2, K); 2) SU(2, C)/∓1 � SO(3, R); 3) H 1 � Spin(3, R). The proof of the isomorphisms 1) and 2) come immediately from the precedent theorem, the C-representation of the the Hamilton quaternion field given in §1, and the isomorphism 3),which we are going to describe in detail(Coxeter [2]). We consider the Hamilton quaternions with reduced norm 1. In which those having zero traces can be identified with the vectors of length 1 of R 3 .
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The Arithmetic of Quaternion Algebra

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