Section 1.1 What is Algebra?
Well, if you ask a random acquaintance “what is algebra?”, more than likely the answer will be something which they learned in high school, though a quick perusal of your textbook may cast a bit of doubt. Still, shouldn’t there be some connection between that subject you learned in high school and the one by that name in College?
In high school, algebra was about solving equations or simple systems of equations, about how to use the quadratic formula to find points of intersection of two conics, and so on. Linear Algebra undertakes the systematic study of solving systems of linear equations which is best answered when cast in terms of algebraic structures called vector spaces, and structure preserving maps between them. Abstract algebra, often called modern algebra, fully embraces the study of sets endowed with one or more binary operations.
One of the main goals of mathematics is to classify things by type, and to characterize when two things are of the same type. For vector spaces being of the same type is called isomorphic, and you learned that two vector spaces are isomorphic if and only if they have the same dimension. That is a remarkably simple characterization. For groups, rings and fields, this characterization is significantly more challenging.
Another important goal of mathematics is to see the manner in which a given object can be built up from simpler ones. For example,
- When is a group \(G\) isomorphic to the direct product of simpler groups?
- How can one classify all the groups \(G\) which contain a normal subgroup isomorphic to a fixed group \(H\) having prescribed quotient \(G/H?\)
Abstract algebra creates an extensive toolbox with which to embark on these investigations.
