College Algebra Exam Review 436

College Algebra Exam Review 436 - 3 3 .2;1/ C 3 3 . Of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 2 P .d;n/ , and suppose that P.³/ holds for all partitions ³ 2 P .d;n/ such that ³ < ² . (This is the inductive hypothesis .) Let p be a homogeneous symmetric polynomial of degree d whose leading term is a ² x ² . As observed previously, p 1 D p ± a ² ´ ² ± is either zero or a homogeneous symmetric polynomial of degree d with leading monomial x ˇ for some ˇ < ² . According to the inductive hypothesis, p 1 is a linear combination of ´ ± ³ with ³ ² ˇ < ² . Therefore, p D p 1 C a ² ´ ² ± is a linear combination of ´ ± ³ with ³ ² ² . 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 ± 3x 2 y ± 3x y 2 ± 3x 2 z ± 6x y z ± 3y 2 z ± 3x z 2 ± 3y z 2 ; p 2 D p 1 C .2;1/ D 3xyz D 3 : Thus, p D ´ 1
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 x 1 ;:::;x n of polynomials that are homo-geneous of total degree d or identically zero is a nitedimensional vector subspace of the K-vector space Kx 1 ;:::;x n . (b) Find the dimension of K d x 1 ;:::;x n . (c) Show that Kx 1 ;:::;x n is the direct sum of K d x 1 ;:::;x n , where d ranges over the nonnegative integers. 9.6.2. Prove Lemma 9.6.1 . 9.6.3. (a) For each d , K d x 1 ;:::;x n is invariant under the action of S n . 1 www.math.uiowa.edu/~ goodman...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online