The Arithmetic of Quaternion Algebra

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2 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD If h /∈ K, its’ minimal polynomial on K is (x − h)(x − h) = x 2 − t(h)x + n(h) The algebra K(h), generated by h over K, is a quadratic extension over K. The reduced trace and the reduced norm of h are simply the images of h by the operations trace and norm on K(h)/K. The conjugation and the identity are the K-automorphism of K(h). Under the usual definition of the trace and the norm of a K-algebra [cf.Bourbaki[1]], the trace of H/K is T = 2t, The norm of H/K is N = 2n. We denote the group of the units in a ring X by X × . Lemma 1.1.1. The invertible elements in H are that with their reduced norm nontrivial. The reduced norm defines a multiplicative homomorphism from H × into K × . The reduced trace is K-linear, and the mapping (h, k) ↦→ t(hk) is a bilinear form non-degenerated on H. Proof. We leave the verification of the following very simple properties to reader as an exercise: n(hk) = n(h)n(k) n(h) �= 0 is equivalent to The invertibility of h , and in this case h −1 = hn(h) −1 , t(ah + bk) = at(h) + bt(k), t(hk) = t(kh), if a, b ∈ K, andh, k ∈ H. The fact that the mapping (h, k) → t(hk) being non-degenerate comes from the assumption that L/K has been separable. In fact, if t(hk) = 0 whatever k ∈ H , we have t(m1m = 0 for every m ∈ L, if h = m1 + m2u,then m1 = 0. Similarly t(m2m = 0 for every m ∈ L, whence m2 = 0 and h = 0. we note that, one of the advantage of the reduced trace is that, in the case of characteristic 2 the trace T = 2t is zero, but the reduced trace is non-degenerate. In case characteristic not 2, we recover the classical definition of the quaternion algebra. the couple {L, θ} is equivalent with a couple {a, b} formed by two nontrivial elements a, b in K and the relations (1) are determine H as the the K-algebra with basis 1, i, j, ij, where the elements i, j ∈ H satisfy i 2 = a, j 2 = b, ij = −ji. (1.3) The transition between(1)and(3) carries out for example by setting L = K(i), θ = b, u = j. Setting k = ij one can write the table of multiplication of i, j, k , which shows these three elements acting symmetrically . The entities in the table are the products hh ′ : h\ h’ t j k i a k -j j -k b i k j -i -ab The conjugation,the reduced trace, and the reduced norm have their expression as follows: if h = x + yi + zj + tk, then h = x − yi − zj − tk, t(h) = 2x, and n(h) = x 2 − ay 2 − bz 2 + abt 2 , the coefficient of k in h should not be confused with the reduced trace. We notice the another important property: the reduced norm defines a quadratic form on the K-vector space V of the subjacent H.
1.1. QUATERNION ALGEBRA 3 We shall see that, the quaternion algebra H is defined by the relations(1) or (3) to the forms {L, θ} or {a, b} when the case is permitted. we also shall consider these notation u, i, j, t(h), h as the standard notation. The fundamental example of a quaternion algebra over K is given by the algebra M(2, k), the matrices of 2 with entities in K. The reduced trace and the reduced norm are the trace and the determinant as the usual sense in M(2, K). It can be identify K with its image in M(2, K) of the K-homomorphism � �which convert a b the unit of K to the identity matrix. Explicitly if h = ∈ M(2, K), � � c d d −b h = , t(h) = a + d, n(h) = ad − bc . We are going to show that −c a M(2, k) satisfies the definition of a quaternion algebra as follows. We choose a matrix with a distinguish value, and set L = K(m). Since m has the same distinguish value as m , it is similar to m.: there exists then an u ∈ GL(2, k) such that umu −1 = m. We verify that t(u) = 0, since t(um) = t(u)m ∈ K for each m ∈ L,from this we deduce u 2 = θ ∈ K ′ . In the following remark we are likely going to explain why is M(2, K) the fundamental example: Over a separably closed field, M(2, K) is the unique quaternion algebra up to an isomorphism. In fact, every separable algebra of dimension 2 over K can not be a field sent by the norm on K × surjectively, and being included in M(2, k)(an inclusion is an injective K-homomorphism). From this we derive that, it is isomorphic to {K + K, 1} � M(2, K) thanks to the realization of M(2, K) as a quaternion algebra done above. tensor product. Let F be a commutative field containing K. We verify directly by the definition the tensor product of a quaternion algebra with F over K is a quaternion algebra over F , and that F ⊗ {L, θ} = {F ⊗ L, θ} . We write the obtained quaternion algebra by HF too. The algebra H is included in HF in a natural way. Taking the separable closure Ks of K as F we see that H is included in M(2, Ks). Definition 1.4. The fields F/K such that HF to be isomorphic to M(2, F ) is called the neutralized fields of H in M(2, k). The inclusions of H in M(2, F ) is called the F -representations. Examples. (1) The quaternion algebra over K has no any K-representation if it is not isomorphic to M(2, K). (2)We define � the�following � matrices: � � � 1 0 0 1 0 1 I = , J = , IJ = . These matrices satisfy the 0 −1 1 0 −1 0 relations (3) with a = b = 1. We derived from them in the case of characteristic unequal to 2, a quaternion algebra {a, b} is isomorphic to � √ √ √ � x + ay b(z + at { √ √ √ |x, y, z, t ∈ K}, b(z − at) x − ay where √ a and √ b are two roots of a and b in Ks . (3) The quaternion field of Hamilton. Historically, the first quaternion algebra ( different from a matrices algebra) was introduced by Hamilton. We denote
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The Arithmetic of Quaternion Algebra

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