of quaternion algebras

The additive identity is (0, 0), the multiplicative identity is (1, 0), and from addition and scalar multiplication of real vectors we have (a, b) = (a, 0) + (0, b) = a(1, 0) + b(0, 1), which looks like a+ bi if we define i to be (0, 1). Real numbers occur as the pairs (a, 0). Hamilton asked himself if it was possible to multiply triples (a, b, c) in a nice way that extends multiplication of complex numbers (a, b) when they are thought of as triples (a, b, 0). In 1843 he discovered a way to multiply in four dimensions, not three, at the cost of abandoning commutativity of multiplication. His construction is called the quaternions. After meeting the quaternions in Section 2, we will see in Section 3 how they can be generalized to a construction called a quaternion algebra. Sections 4 and 5 explore quaternion algebras over fields not of characteristic 2.