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College Algebra Exam Review 255

College Algebra Exam Review 255 - 1 1 x i 2 2 ±±± x i n...

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6.1. A RECOLLECTION OF RINGS 265 In particular, if R has a multiplicative identity element 1 , then .n 1/a D n.1 a/ D na and a.n 1/ D n.a 1/ D na for any n 2 Z and a 2 R . A field is a special sort of ring: Definition 6.1.2. A field is a commutative ring with multiplicative identity element 1 (different from 0 ) in which every nonzero element is a unit . We gave a number of examples of rings and fields in Section 1.11 , which you should review now. There are (at least) four main sources of ring theory: 1. Numbers. The familiar number systems Z , Q , R , and C are rings. In fact, all of them but Z are fields. 2. Polynomial rings in one or several variables. We have discussed poly- nomials in one variable over a field in Section 1.8 . Polynomials in several variables, with coefficients in any commutative ring R with identity ele- ment, have a similar description: Let x 1 ; : : : ; x n be variables, and let I D .i 1 ; : : : ; i n / be a so–called multi-index , namely, a sequence of nonnegative integers of length n . Let x I denote the monomial x I D
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Unformatted text preview: 1 1 x i 2 2 ±±± x i n n . A polynomial in the variables x 1 ;:::;x n with coefficients in R is an expres-sion of the form P I ˛ I x I , where the sum is over multi-indices, the ˛ I are elements of R , and ˛ I D for all but finitely many multi-indices I . Example 6.1.3. 7xyz C 3x 2 yz 2 C 2yz 3 is an element of Q Œx;y;zŁ . The three nonzero terms correspond to the multi-indices .1;1;1;/; .2;1;2/; and .0;1;3/: Polynomials in several variables are added and multiplied according to the following rules: X I ˛ I x I C X I ˇ I x I D X I .˛ I C ˇ I /x I ; and . X I ˛ I x I /. X J ˇ J x J / D X I X J ˛ I ˇ J x I C J D X L ± L x L ; where ± L D X I;J I C J D L ˛ I ˇ J . With these operations, the set RŒx 1 ;:::;x n Ł of polynomials in the vari-ables f x 1 :::;x n g with coefficients in R is a commutative ring with multi-plicative identity....
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