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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 coefcients 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 nitely many multi-indices I . Example 6.1.3. 7xyz C 3x 2 yz 2 C 2yz 3 is an element of Q x;y;z . 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 I x I D X I . I C I /x I ; and . X I I x I /. X J J x J / D X I X J I J x I C J D X L L x L ; where L D X I;J I C J D L I J . With these operations, the set Rx 1 ;:::;x n of polynomials in the vari-ables f x 1 :::;x n g with coefcients in R is a commutative ring with multi-plicative identity....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08