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College Algebra Exam Review 436

# College Algebra Exam Review 436 - 3 ± 3´.2;1 C 3´ 3 Of...

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446 9. FIELD EXTENSIONS – SECOND LOOK Moreover, if p is a symmetric homogeneous polynomial of degree d in n variables whose highest monomial is x , then p is a multiple of m .x 1 ; : : : ; x n / . This shows that P. / holds. Now, fix a partition 2 P .d; n/ , and suppose that P. / holds for all partitions 2 P .d; n/ such that < . (This is the inductive hypothesis .) Let p be a homogeneous symmetric polynomial of degree d whose leading term is a x . As observed previously, p 1 D p a is either zero or a homogeneous symmetric polynomial of degree d with leading monomial x ˇ for some ˇ < . According to the inductive hypothesis, p 1 is a linear combination of with ˇ < . Therefore, p D p 1 C a is a linear combination of with . Example 9.6.9. We illustrate the algorithm by an example. Consider poly- nomials in three variables. Take p D x 3 C y 3 C z 3 . Then p 1 D p 1 3 D 3 x 2 y 3 x y 2 3 x 2 z 6 x y z 3 y 2 z 3 x z 2 3 y z 2 ; p 2 D p 1 C 3 .2;1/ D 3xyz D 3 3 : Thus, p D 1 3 3 .2;1/ C 3 3 . Of course, such computations can be automated. A program in
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Unformatted text preview: 3 ± 3´ .2;1/ C 3´ 3 . Of course, such computations can be automated. A program in Math-ematica for expanding symmetric polynomials in elementary symmetric functions is available on my World Wide Web site. 1 Exercises 9.6 9.6.1. (a) Show that the set K d Œx 1 ;:::;x n Ł of polynomials that are homo-geneous of total degree d or identically zero is a ﬁnite–dimensional vector subspace of the K-vector space KŒx 1 ;:::;x n Ł . (b) Find the dimension of K d Œx 1 ;:::;x n Ł . (c) Show that KŒx 1 ;:::;x n Ł is the direct sum of K d Œx 1 ;:::;x n Ł , where d ranges over the nonnegative integers. 9.6.2. Prove Lemma 9.6.1 . 9.6.3. (a) For each d , K d Œx 1 ;:::;x n Ł is invariant under the action of S n . 1 www.math.uiowa.edu/~ goodman...
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