Chapter 0
Prerequisites
All topics listed in this chapter are covered in
A Primer of Abstract Mathematics
by
Robert B. Ash, MAA 1998.
0.1 Elementary Number Theory
The greatest common divisor of two integers can be found by the Euclidean algorithm,
which is reviewed in the exercises in Section 2.5. Among the important consequences of
the algorithm are the following three results.
0.1.1
If
d
is the greatest common divisor of
a
and
b
, then there are integers
s
and
t
such that
sa
+
tb
=
d
. In particular, if
a
and
b
are relatively prime, there are integers
s
and
t
such
that
sa
+
tb
=1.
0.1.2
If a prime
p
divides a product
a
1
···
a
n
of integers, then
p
divides at least one
a
i
0.1.3 Unique Factorization Theorem
If
a
is an integer, not 0 or
±
1, then
(1)
a
can be written as a product
p
1
···
p
n
of primes.
(2) If
a
=
p
1
···
p
n
=
q
1
···
q
m
, where the
p
i
and
q
j
are prime, then
n
=
m
and, after
renumbering,
p
i
=
±
q
i
for all
i
.
[We allow negative primes, so that, for example,
−
17 is prime. This is consistent with the
general deFnition of prime element in an integral domain; see Section 2.6.]
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CHAPTER 0. PREREQUISITES
0.1.4 The Integers Modulo
m
If
a
and
b
are integers and
m
is a positive integer
≥
2, we write
a
≡
b
mod
m
, and say
that
a
is
congruent
to
b
modulo
m
,i
f
a
−
b
is divisible by
m
. Congruence modulo
m
is an equivalence relation, and the resulting equivalence classes are called
residue classes
mod
m
. Residue classes can be added, subtracted and multiplied consistently by choosing
a representative from each class, performing the appropriate operation, and calculating
the residue class of the result. The collection
Z
m
of residue classes mod
m
forms a
commutative ring under addition and multiplication.
Z
m
is a Feld if and only if
m
is
prime. (The general deFnitions of ring, integral domain and Feld are given in Section 2.1.)
0.1.5
(1) The integer
a
is relatively prime to
m
if and only if
a
is a unit mod
m
, that is,
a
has
a multiplicative inverse mod
m
.
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 Spring '11
 sdd
 Number Theory, Integers, Natural number, Prime number, Greatest common divisor

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