College Algebra Exam Review 434

College Algebra Exam Review 434 - 444 9. FIELD EXTENSIONS...

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Unformatted text preview: 444 9. FIELD EXTENSIONS – SECOND LOOK Define a total order on n-tuples of nonnegative integers and, in particular, on partitions by ˛ > ˇ if the first nonzero difference ˛i ˇi is positive. Note that . / for any permutation . This total order on n-tuples ˛ induces a total order on monomials x ˛ called lexicographic order. Lemma 9.6.7. (a) The leading (i.e., lexicographically highest) monomial of x. (b) X .x1 ; : : : ; xn / D T m .x1 ; : : : ; xn /; is j jDj j where T is a nonnegative integer, T >. D 1, and T D 0 if (c) X mD S .x1 ; : : : ; xn /; j jDj j where S is a nonnegative integer, S >. D 1, and S D 0 if Proof. .x1 ; : : : ; xn / D .x1 x 1 /.x1 D x1 1 x2 2 xn n C x 2/ .x1 x n/ C : : : ; where the omitted monomials are lexicographically lower, and j is the number of k such that k j ; but then j D j , and, therefore, the leading monomial of is x . Since is symmetric, we have D m C a sum of m , where j j D j j and < in lexicographic order. This proves parts (a) and (b). Moreover, a triangular integer matrix with 1’s on the diagonal has an inverse of the same sort. Therefore, (b) implies (c). I Example 9.6.8. Take n D 4 and We have D . 4 /2 . 3 /2 D .4; 4; 3; 3; 2/. Then D .5; 4; 4; 2/. 1 D .x1 x2 x3 x4 /2 .x1 x2 x3 C x1 x3 x4 C x2 x3 x4 /2 .x1 C : : : x4 / 5442 D x1 x2 x3 x4 C : : : ; 5442 where the remaining monomials are less than x1 x2 x3 x4 in lexicographic order. ...
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