Cuaterniones y octoniones

Abstract

The first section of this work discusses algebras. Particularly, the algebras which interest us are division algebras, which are algebras over a field where division is always possible. Then we introduce quaternions. We show that quaternions are a non-commutative division algebra. Also we will see how can we express a quaternion through real and complex matrices of dimensions 2 and 4 respectively, and why the equation $z^2 + 1 = 0$ with $z \in \mathbb{H}$ has infinite solutions. In the last part of this section, we prove the Frobenius Theorem which affirms that the only division algebras of finite dimension over $R$ are the real numbers, the complex numbers and the quaternions. Hamilton discovered quaternions with the idea of using them to study rotations in 3-dimensional space. In the third section of this work we will see how to represent 3-dimensional rotations with unit quaternions. We will introduce the octonions in the fourth part of this work. We will see that octonions form a non-associative division algebra. In the next section we introduce the Cayley-Dickson construction for normed algebras. By this construction, we can obtain the complex numbers from the real numbers, the quaternions from the complex numbers and y the octonions from the quaternions. Finally, we will see that we can define a cross product in $\mathbb{R}^n$ only if $n$ = 1, 3 or 7. We will use this fact to prove a theorem, asserting that the possible dimensions for a normed algebra over $\mathbb{R}$ are only 1, 2, 4, 8. We will deduce from this statement a Theorem of Hurwitz which states that if $n\in\mathbb{N}$, the product of two sums of $n$ squares can be expressed as a sum of $n$ squares only if $n = 1,2, 4, 8$.