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This is clearly a homomorphism of rings if ker f =

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Unformatted text preview: This is clearly a homomorphism of rings. If ker( f ) = { } , then the ring Z is embedded in R , and R has characteristic zero. If ker( f ) = n Z for n > 0, then the ring Z n is embedded in R , and R has characteristic n . 2 43 Chapter 6 Polynomials over Fields Throughout this chapter, K denotes a field, and D denotes the ring K [ T ] of polynomials over K . Like the ring Z , D is an integral domain, and as we shall see, because of the division with remainder property for polynomials, D has many other properties in common with Z as well. Indeed, essentially all the ideas and results from Chapters 1 and 2 carry over almost immediately from Z to D . Recall that for a,b ∈ D , we write b | a if a = bc for some c ∈ D ; note that deg( a ) = deg( b ) + deg( c ). Also, recall that because of the cancellation law for an integral domain, if b | a and b 6 = 0, then the choice of c above is unique, and may be denoted a/b . The units D * of D are precisely the units K * of K ; i.e., the non-zero constants. We call two polynomials a,b ∈ D associates if a = bu for u ∈ K * . Clearly, any non-zero polynomial a is associate to a unique monic polynomial, called the monic associate of a . Note that a polynomial a is a unit if and only if it is associate to 1. Let us call a polynomial normalized if it is either zero or monic. We call a polynomial p irreducible if it is non-constant and all divisors of p are associate to 1 or p . Conversely, we call a polynomial n reducible if it is non-constant and is not irreducible. Equivalently, non-constant n is reducible if and only if there exist polynomials a,b ∈ D of degree strictly less that n such that n = ab . Clearly, if a and b are associate polynomials, then a is irreducible if and only if b is irreducible. The irreducible polynomials play a role similar to that of the prime numbers. Just as it is convenient to work with only positive prime numbers, it is also convenient to restrict attention to monic irreducible polynomials. Corresponding to Theorem 1.2, every non-zero polynomial can be expressed as a unit times a product of monic irreducibles in an essentially unique way: Theorem 6.1 Every non-zero polynomial n can be expressed as n = u · Y p p ν p ( n ) , where u is a unit, and the product is over all monic irreducible polynomials, with all but a finite number of the exponents zero. Moreover, the exponents and the unit are uniquely determined by n . To prove this theorem, we may assume that n is monic, since the non-monic case trivially reduces to the monic case. 44 The proof of the existence part of Theorem 6.1 is just as for Theorem 1.2. If n is 1 or a monic irreducible, we are done. Otherwise, there exist a,b ∈ D of degree strictly less than n such that n = ab , and again, we may assume that a and b are monic. We then apply an inductive argument with a and b ....
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