Elementary Abstract Algebra (Clark)

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This book is intended for a one semester introduction to abstract algebra. We assume that students have some familiarity with basic set theory linear algebra and calculus. But very little of this nature will be needed To a great extent the course is self contained except for the requirement of a certain amount of mathematical maturity. And hopefully the student’s level of mathematical maturity will increase as the course progresses.

The quaternions were invented by Sir William Rowan Hamilton about 1850. Hamilton was perhaps the first to note that complex numbers could be thought of as a way to multiply points in the plane. He then had the idea of trying to find a way to multiply points in so that the field axioms would be satisfied. He was unable to do this, but he finally found a way to define multiplication on so that the multiplication together with ordinary vector addition of elements of would satisfy all the field axioms except for commutativity of multiplication. He called these new objects quaternions. They turned out, like complex numbers, to have many applications in engineering and physics. This “number system” is denoted by for Hamilton since is already taken to denote the rational numbers.