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 expression of the form P I ˛ I x I , where the sum is over multiindices, the ˛ I are elements of R , and ˛ I D for all but ﬁnitely many multiindices 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 multiindices .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 variables f x 1 :::;x n g with coefﬁcients in R is a commutative ring with multiplicative identity....
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 Fall '08
 EVERAGE
 Algebra, Ring, Prime number, Commutative ring

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