ABSTRACT
In Chapter 2 we proved that every integer ( 6= 0,±1) can be written as a product of irreducible integers, and this decomposition is essentially unique. These irreducible integers turn out to be those integers that we call primes. To summarize, in that chapter we proved the following important theorems:
• The Division Theorem for integers (Theorem 2.1), • Euclid’s Algorithm (which yields the gcd of two integers) (Theorem 2.3), • The GCD identity that gcd(a, b) = ax+ by, for some integers x and y (Theorem 2.4), • Each non-zero integer ( 6= ±1) is either irreducible or a product of irreducibles (Theo-
rem 2.8),
• An integer p is irreducible if and only if p is prime (that is, if p|ab, then either p|a or p|b) (Theorem 2.7), and • Each non-zero integer (6= ±1) is uniquely (up to order and factors of −1) the product
of primes (Theorem 2.9).
