The Arithmetic of Quaternion Algebra

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12 CHAPTER 1. QUATERNION ALGEBRA OVER A FIELD Theorem 1.3.7. (The finite Group of real quaternion) The finite subgroup of H × are conjugate to the following groups: (1) the cyclic groups of order n generated by sn = cos(2π/n) + i sin(2π/n). (2)the group of order 4n generated by s2n and j called dicyclic. (3)the group of order 24, called the bilinear tetrahedral group, . E24 = {±1, ±i, ±j, ±ij, ±1 ± i ± j ± ij } 2 (4)the group of order 48, called the bilinear octahedral group, 1 − E24 ∪ 2 2 x, where x = all the sums or differences of the possible two distinct elements among 1, i, j, ij. (5) the group of order 120, called the bilinear icosahedral group, E120 = E24 ∪ 2 −1 x,,x = all the product of an element of E24 and +i, +τj, +τ −1 ij, where τ = ( √ 5 + 1)/2. We obtain by the same reason the description of all the possible finite groups of quaternion algebra embedded in H, i.e. such that the center K is embedded in R, and HR = H. Generators and relations(Coxeter [2],pp.67-68). Let (p, q, r) be the group defined By x p = y q = z r = xyz = 1. the group is finite for (2, 2, n), (2, 3, 3), (2, 3, 4), (2, 3, 5) and isomorphic to the rotation groups Dn, A4, S4, A5. Using the correspondence 1 ↔ 2 given by the mapping (cos α + r sin α) → (r, 2α) which was defined in the proof of theorem 3.4, we see that the group < p, q, r > defined by x p = y q = z r = xyz = u, u 2 = 1 has the following special cases: the dicyclic group < 2, 2, n > of order 4n , the bilinear tetrahedral group < 2, 3, 3 > of order 24, the bilinear octahedral group < 2, 3, 4 > of order 48, the bilinear icosahedral group < 2, 3, 5 > of order 120. Proposition 1.3.8. (Classical isomorphisms). The bilinear tetrahedral group is isomorphic to the group SL(2, F3). The bilinear icosahedral group is isomorphic to the group SL(2, F5) We shall prove this isomorphism in Ch.V,§3. Exercises 1. The isotropic group of the finite quaternion group)(Vigneras[3]). Let K is a finite extension of Q and H/K is a quaternion algebra having an inclusion in H. The group H × acts on H 1 by inner automorphism. Prove
1.4. ORDERS AND IDEALS 13 the isotropic groups in H × of the finite subgroup of H 1 are given in the following table: group isotropic group cyclic < sn >, n > 2 < K(sn) × , tn > where tnsn = s −1 n tn, tn ∈ H × dicyclic < s2n, j?, n ≤ 2 < s2n, j, 1 + s2n, K × > binary tetrahedral E24, or < i, j > < K × E24, 1 + i > binary octahedral E48 K × E48 binary icosahedral E120 K × E120 2. The order of the elements of the finite quaternion group. 1)Prove the elements in quaternion group of order 4n which are generated by s 2n and j has the form s t 2nj, where 0 ≤ t ≤ 2n−1 is always of order 4. 2)Find in the bilinear groups E24, E48, E120 the number of elements with a given order( it suffices to notice that the elements with reduced trace 0, respectively ,−1, 1, ± √ 2.τor − τ −1 , τ −1 or − τ, are of order 4, respectively, 3, 6, 8, 5, 10). 3)Deduce from 2) that the bilinear octahedral group E48 is not an isomorphic to the group GL(2, F3) of order 48. 1 3. a characterization of the quaternion fields (Van Praag[1]). prove, if H is a quaternion field with its center a commutative field K, then the set consisting of 0 and the elements x ∈ H, x 2 ∈ K, butx /∈ K is an additive group. Conversely, if H is a field of characteristic different from 2, such that the set mentioned above is a nonzero additive group, then H is a quaternion field. 4. The rotations of H (Dieudonné [2] or Bourbaki [3]). A rotation of H is a proper isometry of the subjacent quaternion space of H. prove every rotation of H is the mapping of the form: ua,b : x → axb , where a, b ar two quaternions such that n(a)n(b) �= 0. Show that two notations ua,b and uc,d are equal if and only if a = kc, b = k −1 d, where k is a nonzero element of the center of H (suppose the characteristic is not 2). 1.4 Orders and ideals The aim of this section is to give some definitions based on the orders, the ideals, and the reduced discriminants, which will be used in the subsequent chapters, where K is a local or global field. Our purpose is not to reestablish a frame of a theory on a Dedekind ring, but only make some definitions more precise, and these definitions will be adopted in sequel. For a more complete exposition one can consult the book of Reiner [1] or Deuring [1]. The notations used here are standard in the following chapters. Let R be a Dedkind ring, i.e. a noetherian, integrally closed (hence integral) ring such that all of its nonzero prime ideals are maximal. Examples: Z, Z[ 1 p ] for prime p, Z[i] and generally the integer ring of a local or 1 This remark is made by Daniel Perrin friendly.
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The Arithmetic of Quaternion Algebra

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