MATH
notes

Theorem 57 if k is field then for a b k t with b 6 0

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Theorem 5.7 If K is field, then for a, b K [ T ] with b 6 = 0 K , there exist unique q, r K [ T ] such that a = bq + r and deg( r ) < deg( b ) . Proof. Clear. 2 39

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Theorem 5.8 For a ring R and a R [ T ] and x R , a ( x ) = 0 R if and only if ( T - x ) divides a . Proof. Let us write a = ( T - x ) q + r , with q, r R [ T ] and deg( r ) < 1, which means that r R . Then we have a ( x ) = ( x - x ) q ( x ) + r = r . Thus, a ( x ) = 0 if and only if T - x divides a . 2 With R, a, x as in the above theorem, we say that x is a root of a if a ( x ) = 0 R . Theorem 5.9 Let D be an integral domain, and let a D [ T ] , with deg( a ) = k 0 . Then a has at most k roots. Proof. We can prove this by induction. If k = 0, this means that a is a non-zero element of D , and so it clearly has no roots. Now suppose that k > 0. If a has no roots, we are done, so suppose that a has a root x . Then we can write a = q ( T - x ), where deg( q ) = k - 1. Now, for any root y of a with y 6 = x , we have 0 D = a ( y ) = q ( y )( y - x ), and using the fact that D is an integral domain, we must have q ( y ) = 0. Thus, the only roots of a are x and the roots of q . By induction, q has at most k - 1 roots, and hence a has at most k roots. 2 Theorem 5.10 Let D be an infinite integral domain, and let a D [ T ] . If a ( x ) = 0 D for all x D , then a = 0 D . Proof. Exercise. 2 With this last theorem, one sees that for an infinite integral domain D , there is a one-to-one correspondence between polynomials over D and polynomial functions on D . 5.3 Ideals and Quotient Rings Throughout this section, let R denote a ring. Definition 5.11 An ideal of R is a additive subgroup I of R that is closed under multiplication by element of R , that is, for all x I and a R , xa I . Clearly, { 0 } and R are ideals of R . Example 5.12 For m Z , the set m Z is not only an additive subgroup of Z , it is also an ideal of the ring Z . 2 Example 5.13 For m Z , the set m Z n is not only an additive subgroup of Z n , it is also an ideal of the ring Z n . 2 If d 1 , . . . , d k R , then the set d 1 R 1 + · · · + d k R := { d 1 a 1 + · · · + d k a k : a 1 , . . . , a k R } is clearly an ideal, and contains d 1 , . . . , d k . It is called the ideal generated by d 1 , . . . , d k . Clearly, any ideal I that contains d 1 , . . . , d k must contain d 1 R 1 + · · · + d k R . If an ideal I is equal to dR for some d R , then we say that I is a principal ideal . Note that if I and J are ideals, then so are I + J := { x + y : x I, y J } and I J . 40
Throughout the rest of this section, I denotes an ideal of R . Since I is an additive subgroup, we may adopt the congruence notation in § 4.3, writing a b (mod I ) if and only if a - b I . Note that if I = dR , then a b (mod I ) if and only if d | ( a - b ), and as a matter of notation, we may simply write this congruence as a b (mod d ). If we just consider R as an additive group, then as we saw in § 4.3, we can form the additive group R/I of cosets, where ( a + I ) + ( b + I ) := ( a + b ) + I . By considering also the multiplicative structure of R , we can also view R/I as a ring. To do this, we need the following fact.

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• Spring '13
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