Get Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 PDF By Frederick M. Goodman

ISBN-10: 0979914205

ISBN-13: 9780979914201

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Additional resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)

Example text

So we will consider 1 to have a prime factorization as well; it is the product of an empty collection of primes. A fundamental question is whether there exist infinitely many prime numbers or only finitely many. C. 6. There are infinitely many prime numbers. Proof. We show than for all natural numbers n, there exist at least n prime numbers. There exists at least one prime number, because 2 is prime. Let k be a natural number and suppose that there exist at least k prime numbers. We will show that there exist at least k C 1 prime numbers.

7. 7. Œb C Œc/ D ŒaŒb C ŒaŒc: Multiplication in Zn has features that you might not expect. On the one hand, nonzero elements can sometimes have a zero product. For example, in Z6 , Œ4Œ3 D Œ12 D Œ0. We call a nonzero element Œa a zero divisor if there exists a nonzero element Œb such that ŒaŒb D Œ0. Thus, in Z6 , Œ4 and Œ3 are zero divisors. On the other hand, many elements have multiplicative inverses; an element Œa is said to have a multiplicative inverse or to be invertible if there exists an element Œb such that ŒaŒb D Œ1.

Proof. 9. ■ This completes our treatment of unique factorization of polynomials. Before we leave the topic, let us notice that you haven’t yet learned any general methods for recognizing irreducible polynomials, or for carrying out the factorization of a polynomial by irreducible polynomials. In the integers, you could, at least in principle, test whether a number n is prime, and find its prime factorsp if it is composite, by searching for divisors among the natural numbers Ä n. For an infinite field such as Q, we cannot 54 1.