By Frederick M. Goodman

**Read Online or Download Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) PDF**

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**Extra info for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) **

**Example text**

21. The factorization of a polynomial in KŒx into irreducible factors is essentially unique. That is, the irreducible factors appearing are unique up to multiplication by nonzero elements in K. 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.

The fundamental fact about divisibility is the following familiar result (division with remainder): It is always possible to divide an integer a by any divisor d 1, to get a quotient q and a remainder r, with the remainder being strictly smaller than the divisor. I would like to emphasize how the proof of this fact is related to the algorithm for long division which you learned in school. How does this algorithm work (for a a positive number)? In case a < d , we just put q D 0 and r D a. If a d , first we guess an approximation q1 for the quotient such that 0 < q1 d Ä a, and compute r1 D a q1 d , which is less than a.

Every integer belongs to exactly one of these two classes. a/ D P if a 0). If n 2 N and p 2 P , then, depending on the choice of n and p, the sum n C p can be in either of N or P ; that is, the sum of a positive number and a negative number can be either positive or negative. n C p/. Once we have cleared the hurdle of defining sensible operations on Zn , it is easy to check that these operations satisfy most of the usual rules of arithmetic, as recorded in the following proposition. 1). For example, the commutativity of multiplication on Zn is shown as follows.