Modern Algebra 1: Overview

What is Modern Algebra (also referred to as Abstract Algebra) about? To begin to understand this, you need to think about your first experiences with algebra. At that time, you had spent years studying operations on numbers of various sorts (whole numbers, fractions, decimals). The new idea you learned about in algebra was operating on unknown numbers, which are now familiar to you as variables. In essence, you learned to do arithmetic operations on variables without knowing (or needing to know) what specific numbers those variables represented. In modern algebra we carry this abstraction a step further.

By now, you have seen that there are many different number systems that are useful in different contexts: integers, rational numbers, real numbers, vectors, matrices, and so on. We use algebraic notation and operation symbols in all of these systems that looks essentially the same. For example, without knowing a context, an equation such as

Ax + By = 0

might have many different meanings. Maybe A and B are matrices and x and y are vectors. Or maybe A and B are real constants and x and y are vectors. Or all the variables could represent real numbers, or integers, or complex numbers.

As you also have seen, different number systems (or more properly, algebraic systems) have somewhat different properties. If you are talking about real numbers, then AB = BA is always true. But not if you are talking about matrices. Similarly, when dealing with equations, x⁴ + 5 = 0 has no solutions if you are considering integers, two solutions if you are considering real numbers, and four solutions if you are considering complex numbers.

In Modern Algebra we study properties of number systems in the abstract -- that is, without knowing specificially what those systems are. Just as regular algebra allowed you to perform operations in which the exact nature of the numbers is left unknown, so in modern algebra we allow the exact nature of the operations to be unknown.

If you have studied linear algebra, you have already seen this sort of idea in action. In linear algebra, the algebraic properties necessary for posing and solving linear systems can be formulated in a set of around 10 axioms or assumptions. Any algebraic system that satisfies these axioms is called a vector space, and there are many different sorts of vector spaces, made up of matrices, or columns of numbers, or polynomials, or functions, to name a few. Every vector space has common properties that can be proved using the 10 axioms. For example, in any vector spacee there is a notion of linear independence, and of basis, and dimension. In any vector space, the concepts of basis and dimension govern the sorts of solutions that are possible for a linear system of equations.

Indeed, linear algebra is properly a subtopic of modern algebra. But we will be studying other sorts of algebraic systems, with different sets of axioms, besides those for vector spaces.

What is modern algebra good for? The short answer is that this is an incredibly powerful and indispensible tool with applications throughout mathematics. It shows up in a central way in great theorems of number theory (like Fermat's last theorem) and applications to encryption and coding. In our course, we will see how modern algebra leads to important results in two classical areas of mathematics: solving polynomial equations and making geometric constructions. To give a preview, here are a couple of major results that puzzled mathematicians for centuries, and can be derived using modern algebra:

1. There is no compass-and-straightedge algorithm for trisecting angles.

2. There is no formula using squareroots, cuberoots, and so on, for solving polynomial equations of degree 5.

The first of these we will see toward the end of our course, or early in the second half of the course. The second is a feasible goal for a modern algebra course, although we may not be able to reach it before the end of the year.