Chapter 4
Abstract Algebra
While an oversimplification,abstract algebragrew out of an attempt
to solve and otherwise understand polynomial equations (or systems of
polynomial equations). A relative high point can be found in the early
nineteenth century with E. Galois’ proof that polynomial equations
of degree at least 5 need not be solvable by the “usual” processes of
addition, subtraction, multiplication, division, and extraction of roots
as applied to the polynomial’s coefficients. What’s remarkable is not so
much the result itself but rather the methods employed. This marked
the beginning of a new enterprise, now calledgroup theorywhich soon
took on a life of itself, quite apart from playing a role in polynomial
equations.
The language and level of abstraction in group theory quickly be-
gan to spread, leading to the somewhat larger discipline ofabstract
algebra. We’ll attempt to give the serious student a meaningful intro-
duction in this chapter.
4.1 Basics of Set Theory
In this section we shall consider some elementary concepts related to
setsand theirelements, assuming that at a certain level, the students
have encountered the notions. In particular we wish to review (not
necessarily in this order)
- Elementcontainment(∈)
- Containmentrelationships between sets (⊆,⊇,⊂,(same as
(),⊃,(same as )))
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