Introduction to Abstract Algebra

Alexander Paulin

If you ask someone on the street this question, the most likely response will be: “Something horrible to do with x, y, and z.” If you’re lucky enough to bump into a mathematician, you might get something like: “Algebra is the abstract encapsulation of our intuition for composition.”

By composition, we mean the concept of two objects coming together to form a new one, for example, adding two numbers or composing real-valued single variable functions. As we shall discover, the seemly simple idea of composition hides vast hidden depth. Algebra permeates all of our mathematical intuitions, and the first mathematical concepts we ever encounter are the foundation of the subject.