The Arithmetic of Quaternion Algebra

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22 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD (f) From it deduce that if P (x) has only a finite number of roots, the number is less than or equal to the degree of P . (g) Suppose H to be the field H of Hamilton quaternion. Prove that if P (x) is not the polynomial 1, then P (x) = 0 always has a root in H, and has an infinite number of roots if and only if P (x) is divided by an irreducible polynomial with real coefficients of degree 2. (h) Let h1, h2, ..., hr be the elements in H, but not belonging toK and not conjugate pairwise, and m1, m2, ..., mr be the integers being greater than or equal to 1. We say that h is a root of P (x) of multiplicity m if P m m+1 h divides n(P ), and Ph does not divide n(P ). Show if every mi equals 1, there exists a unique monic polynomial P (x) , its roots are only these quaternions hi (1 ≤ i ≤ r) with multiplicity mi, and the degree of P (x) equals m = � mi. If not the case, Prove there exists an infinitely many monic polynomials of degree m with this property.
Chapter 2 Quaternion algebra over a local field In the chapter, K is a local field, in other words, a finite extension K/K ′ of a field K ′ called its prime subfield1 are equal to one of the following fields: – R ,the field of real numbers, – Qp, the field of p-adic numbers, – Fp[[T ]], the field of formal series of one indeterminate over the finite field Fp. The fields R, C are called archimedean, the field K �= R, C are called non-archimedean. If K ′ �= R . Let R be the integral ring of K and π, k = R/πR be the uniform parameter and residue field respectively. We use Lnr to denote the unique quadratic extension of K in the separable closure Ks of K which is non ramified, i.e. satisfies one of the following equivalent properties: (1) π is a uniform parameter of Lnr, (2) R × = n(R × L ) where RL is the integral ring of Lnr, (3) [kL : k] = 2, where kL is the residue field of Lnr. Let H/K be a quaternion algebra. All of the notions about the orders and the ideals in H are relative to R. 2.1 Classification The following theorem provides a very simple classification of the quaternion algebra over a field. Theorem 2.1.1. (Classification). Over a local field K �= C there exists a quaternion field uniquely determined up to an isomorphism. We have seen in ch.1,§1 that M(2, K) is the unique quaternion algebra over C up to isomorphism. The theorem of Frobenius implies this theorem in the case of K = R. Before giving the proof of the theorem we give some application of it. Definition 2.1. We shall define an isomorphism ε of Quat(K) to {∓1} by setting for a quaternion algebra H/K, ε(H) = −1 if H is a field, ε(H) = 1 1 This notation for the prime subfield is not the usual one, but it will be used in the sequel. 23
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The Arithmetic of Quaternion Algebra

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