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Quaternion algebras and quadratic forms By Wedderburn's theorem (Theorem 2.2.6) and the fact that a central division algebra over F must have square dimension, we can explicitly list the types of simple F-algebras in small dimensions. In 1-dimension, there is just F. In 2- and 3-dimensions, there are just quadratic and cubic fields (exercise below). QUAINT Chapter 6: Arithmetic of quaternion algebras Kimball Martin 6.1.1 Local orders In this section, let F be a p-adic field and B be a quaternion algebra over F. Then, up to isomorphism, either B = D (nonsplit) or B = M2(F) (split), where D denotes the unique quaternion division algebra over F. Here we summarize the theory of (a large class of) quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo-metric meaning is also more obvious as the rotation axis and angle can be trivially recovered. The quaternion algebra to be introduced will also allow us to easily compose rotations. This is because their cross product as vectors. For any quaternion algebra B = (a,b) F, describe a dot product and cross product on F3 which is related to multiplication in B0. (Formulas for these new products will depend on a and b.) Quaternion algebras over Q 22. Suppose a,b ∈ R× are not both negative. Show (a,b) Q becomes a subring of M 2(R) by 1 7→ 1 A quaternion algebra ( a, b) F is either a division algebra or isomorphic to the matrix algebra of 2 × 2 matrices over F; the latter case is termed split. [4] The norm form. defines a structure of division algebra if and only if the norm is an anisotropic quadratic form, that is, zero only on the zero element. The conic C ( a, b) defined by. with x for every d ∈ D, so L can't be onto (D is non-commutative). Use linear algebra to explain why L(d) = 0 for some non-zero d. 7. Let B be a quaternion algebra over a field F and let f: B → B be an F-algebra homomorphism. Then f(q) = f(q) for every q ∈ B. (Hint: Reduce to the case of pure quaternions q and show f(B0) ⊂ B0.) 8. El conjunto de los cuaterniones puede expresarse como: o equivalentemente: Entonces un cuaternión es un número de la forma a + bi + cj + dk, donde a, b, c, y d son números reales unívocamente determinados por cada cuaternión. Análogamente, un cuaternión puede expresarse como el producto interno ( componente a componente) de dos vectores Quaternion algebra means something more general than the algebra of Hamilton's quaternions (for which there exists already a Maple package) . Let we recall some definitions. A quaternion algebra over is a central simple algebra of dimension 4 over . To give a quaternion algebra is equivalent to give a pair of non-zero rational numbers so that The quaternion algebras (a;b F) and (a0;b0 F) are isomor-phic as algebras if and only if their norm forms are isometric as quadratic forms. One way around this implication is clear since any isomorphism of the algebras will give an isometry of the norm forms. For the converse first observe that an isometry of norm forms is not in general Theorem: A quaternion algebra that is not a division ring is isomorphic to M 2(F). Proof We already know that all quaternion algebras of the form (a;c2) F and by isomor-phism those of the for