文档介绍：page 1 of Enrichment
Enrichment
Chapters 1-4 form an idealized undergraduate course, written in the style of a graduate
text. To help those seeing abstract algebra for the first time, Ihave prepared this section,
which contains advice, explanations and additional examples for each section in the first
four chapters.
Section
When we say that the rational numbers form a group under addition, we mean that
rational numbers can be added and subtracted, and the result will inevitably be rational.
Similarly for the integers, the real numbers, and plex numbers. But the integers
(even the nonzero integers) do not form a group under multiplication. If a is an integer
other than ±1, there is no integer b such that ab = 1. The nonzero rational numbers form
a group under multiplication, as do the nonzero reals and the plex numbers.
Not only can we add and subtract rationals, we can multiply and divide them (if the divisor
is nonzero). The rational, reals plex numbers are examples of fields, which will be
studied systematically in Chapter 3.
Here is what the generalized associative law is saying. pute the product of the
elements a, b, c, d and e, one way is to pute bc, then (bc)d, then a((bc)d), and finally
[a((bc)d)e]. Another way is (ab), then (cd), then (ab)(cd), and finally ([(ab)(cd)]e). All
procedures give the same result, which can therefore be written as abcde.