Math 373/578, Spring 2013
Ch 1.4
Section 1.4: Prime Numbers, Unique Factorization, and Finite Fields (p26)
(1.4A) In modular arithmetic, we can add, subtract, and multiply integers mod m, but we can only
divide by a number a in Z m if gcd(a,m)=1. Thus, if m is prime, we can divide by every nonzero
element of Z m .
(1.4B) Def: An integer p is called a prime if p ≥ 2 and if the only positive integers dividing p are 1
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and p.
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(1.4C) Proposition 1.20: Let p be a prime number, and suppose that p divides the product ab of two integers
a and b. Then p divides at least one of a€and b. In general, if p divides a product of integers, say p|a1a 2 an
then p divides at least one of the individual a i .
(1.4D) Theorem 1.21: (Fundamental Theorem of Arithmetic) Let a ≥ 2 be an integer. Then a can be
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factored as a product of prime numbers: a = p1e1 p 2e 2  pre r . Further, other than rearranging the order of the
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primes, this factorization into prime powers is unique.
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(1.4E) Def: In the factorization of a positive integer a into primes, each prime p appears to a
particular power, which we€denote ord p (a) and call the order (or exponent) of p in a.
• We set ord p (1) = 0 for all primes.
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(1.4F) Ex: The factorization of 2592 is 2592 = 25 ⋅ 3 4 , so ord 2 (2592) = 5 and ord 3 (2592) = 4 .
(1.4G) €
Proposition 1.22: Let p be a prime. Then every nonzero element a in Z p has a multiplicative
−1
inverse, that is, there is a number €
b satisfying ab ≡1(mod
p) . We denote this
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€ value of b by a (mod p) .
(1.4H) Recall: The extended Euclidean algorithm states that the equation au + bv = gcd(a,b) always
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has a solution (a,b,u,v integers). So, we can calculate a −1 (mod p) by solving the equation au + pv =1
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for u and v, and u will be the multiplicative inverse of a mod p.
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(1.4I) Another way to state Prop 1.22: if p is prime, then (Z p )* = {1,2,..., p −1} . In other words, the
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set Z p − {0} consists of units and is closed under multiplication.
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(1.4J) Def: If p is prime, then the set Z p of integers
€ mod p with its addition, subtraction,
multiplication, and division rules is an example of a field. (In particular, a finite field.)
• A field is a general name for a (commutative) ring in which every nonzero element has a
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multiplicative inverse.
• Examples of fields: R = real numbers; Q = rational numbers; C = complex numbers
Exercises: 1.28
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