In mathematics, we often encounter objects that are set with various operations that can be performed on them. For instance, one may add and multiply integers and do the same with rational numbers, real numbers, and even complex (or imaginary) numbers. Or, given two functions that input and output real numbers, we can compose them and we can add vectors or multiply them by scalars.
In abstract algebra, we attempt to provide lists of properties that common mathematical ob- jects satisfy. Given such a list of properties, we impose them as “axioms,” and we study the objects’ properties that satisfy them. The objects we deal with most in the first part of these notes are called groups, rings, and fields.