Mathematics begins with the integers. The branch of mathematics known as
algebra devotes itself to studying mathematical systems that are, in some sense,
generalizations of the integers or closely related gadgets. This chapter gathers
together some of the important properties of the integers and modular arithmetic
that will be used throughout the rest of the book.
1. Integers
1.1. Basic Properties. The set Z of integers is the well-loved set
Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4,...}
that most of us learned about in grade school. Instead of beating around the bush,
we assume the reader is familiar with the elementary properties of addition and
multiplication in Z. At the risk of the reader rolling his or her eyes, we list some
of the basic properties of Z since our later generalizations closely mimic them.
Axiom 1 (Additive and Multiplicative Properties). Let a, b, c Z.
The properties of addition are:
(1) Associativity: (a + b) + c = a + (b + c).
(2) Commutativity: a + b = b + a.
(3) Identity: 0 + a = a.
(4) Inverses: a + (−a) = 0.
The properties of multiplication are:
(1) Associativity: (ab) c = a (bc).
(2) Commutativity: ab = ba.
(3) Identity: 1 · a = a.
There is a law relating addition and multiplication:
(1) Distributivity: a (b + c) = ab + ac.
There is a zero divisor law:
(1) ab = 0 if and only if a = 0 or b = 0.
We also assume that the reader is familiar with the standard elementary prop-
erties of inequalities, which we do not even bother to list. However, a related notion
called the Axiom of Well-Ordering is worth mentioning. Depending on how one
sets up the integers, the Axiom of Well-Ordering can be viewed as a straight-up
axiom or as a corollary of the properties of positive integers, which we also do not
bother to explicitly list. Instead, recall that a subset S Z is bounded above
if there exists N Z so that a N whenever a S. For example, the set
S = {..., −2, −1, 0, 1, 2, 3} is bounded above (by, say, 3 or 4 or 5) while the set
E = {..., −2, 0, 2, 4, 6,...} of even integers is not bounded above. Notice the set
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