K i 0 a i t i is zero if and only if a i 0 r for 0 i

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k i =0 a i T i is zero if and only if a i = 0 R for 0 i k , and two polynomials are equal if and only if their difference is zero. This distinction is important, since there are rings R over which two different polynomials define the same function. One can of course define the ring of polynomial functions on R , but in general, that ring has a different structure from the ring of polynomials over R . Example 5.11 In the ring Z p , for prime p , we have x p - x = [0] for all x Z p . But consider the polynomial a = T p - T Z p [ T ] . We have a ( x ) = 0 R for all x 0 R , and hence the function defined by a is the zero function, yet a is not the zero polynomial. 2 5.2.2 Basic properties of polynomial rings Let R be a ring. 38
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For non-zero a R [ T ], if a = k i =0 a i T i with a k 6 = 0 R , we call k the degree of a , denoted deg( a ), and we call a k the leading coefficient of a , denoted lc( a ), and we call a 0 the constant term of a . If lc( a ) = 1 R , then a is called monic . Note that if a, b R [ T ], both non-zero, and their leading coefficients are not both zero divisors, then the product ab is non-zero and deg( ab ) = deg( a ) + deg( b ). However, if the leading coefficients of a and b are both zero divisors, then we could get some “collapsing”: we could have ab = 0 R , or ab 6 = 0 R but deg( ab ) < deg( a ) + deg( b ). For the zero polynomial, we establish the following conventions: its leading coefficient and constant term are defined to be 0 R , and its degree is defined to be “ -∞ ”, where it is understood that for all integers x Z , -∞ < x , and ( -∞ ) + x = x + ( -∞ ) = -∞ , and ( -∞ ) + ( -∞ ) = -∞ . This notion of “negative infinity” should not be construed as a useful algebraic notion — it is simply a convenience of notation; for example, it allows us to succinctly state that for all a, b R [ T ], deg( ab ) deg( a ) + deg( b ), with equality holding if the leading coefficients of a and b are not both zero divisors. Theorem 5.5 Let D be an integral domain. Then 1. for all a, b D [ T ] , deg( ab ) = deg( a ) + deg( b ) ; 2. D [ T ] is an integral domain; 3. ( D [ T ]) * = D * . Proof. Exercise. 2 5.2.3 Division with remainder An extremely important property of polynomials is a division with remainder property, analogous to that for the integers: Theorem 5.6 (Division with Remainder Property) Let R be a ring. For a, b R [ T ] with lc( b ) R * , there exist unique q, r R [ T ] such that a = bq + r and deg( r ) < deg( b ) . Proof. Consider the set S of polynomials of the form a - xb with x R [ T ]. Let r = a - qb be an element of S of minimum degree. We must have deg( r ) < deg( b ), since otherwise, we would have r 0 := r - (lc( r )lc( b ) - 1 T deg( r ) - deg( b ) ) · b S , and deg( r 0 ) < deg( r ), contradicting the minimality of deg( r ). That proves the existence of r and q . For uniqueness, suppose that a = bq + r and a = bq 0 + r 0 , where deg( r ) < deg( b ) and deg( r 0 ) < deg( b ). This implies r 0 - r = b ( q - q 0 ) . However, if q 6 = q 0 , then deg( b ) > deg( r 0 - r ) = deg( b ( q - q 0 )) = deg( b ) + deg( q - q 0 ) deg( b ) , which is impossible. Therefore, we must have q = q 0 , and hence r = r 0 . 2 If a = bq + r as in the above theorem, we define a rem b := r .
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