Math Placement Advice/What is ...?/What is Algebra?

The importance of 401 and 405 to majors cannot be overstated. This course and 405 are the only two courses absolutely required for all majors. These two together provide an introduction to the ideas and the methods of modern abstract mathematics. In 401 and 405 you will learn algebra and analysis, but you will also learn how proofs are developed and used, and methods to construct your own proofs. This involves an important transition in mathematical maturity: Until this point you have focused on solving problems by applying methods, and the central unit of learning is the "new method." Once you reach these more abstract courses, you will focus on uncovering information by applying theorems and their proofs. The central unit of learning will be the theorem. This is somehow more natural and more straightforward, because "theorem" is just another word for "fact." Instead of learning to do things you will now learn facts, and infer how to do things. In the best case, this transition happens early, even in Calculus class. In any case, expect it to be complete after you do 401 and 405!

Wondering what you'll be getting into? Algebra I will teach you about groups, rings, and fields, but of course you don't already know what groups, rings, and fields are. So what is algebra? Roughly speaking, our system of numbers has objects (3,0,e,pi), operations (+,-,*,/), and rules (a+b=b+a, a(b+c)=ab+ac, etc.). In algebra, we take away the objects, leaving operations, operating abstractly on mysterious unknowns, and rules, reassuring us of some remaining structure. We then ask "What do we still know, despite self-imposed ignorance of objects?" The answer, invariably, is "quite a lot," which by the way means ultimately that most of our knowledge of numbers comes not directly from our great understanding of the numbers themselves, but from the operations and rules governing them. Eventually, we can "fill the void," substituting other objects (matrices, polynomials, spatial rotations, remainders-by-seven, possible rearrangements of five chairs, etc.) in place of numbers, and produce pseudo-numerical knowledge about structures which are not at all numerical in nature.

Groups, rings, and fields, are names for instantiations of these rules with various levels of requirements that make them more and more like the real numbers. Groups have the fewest requirements, and include lots of crazy structures: remainders-by-seven, the Rubik's cube configuration space, the set of square matrices of nonzero determinant, the real number system, the set of spatial rotations, and elliptic curves. Fields are the most like the normal number system, but can still be strange: There is a field with sixteen elements in which + and - are two names for the same operation!

What is algebra?