Algebra/Chapter 10/Symmetric Polynomials/Fundamental Theorem of Symmetric Polynomials

0 95% #FFFFFF Fundamental Theorem of Symmetric Polynomials: Let be a field, and let be a symmetric polynomial. Then can be expressed uniquely as a polynomial , such that: 's degree does not exceed 's degree. If has integer coefficients, then so does . Proof First, we shall describe an algorithm for finding the desired polynomial . Let us define initial conditions and . Find such that . Define . Write . If , return to step 1 and began the process over with the index .If , move on to step 5. Write . In order to prove the algorithm we need two lemmas. Lemma 1: The leading monomial in satisfies . Proof: Let us assume there exists an index such that . Then there exists a permutation such that But the polynomial contains the monomial , which is of higher order than . A contradiction. Lemma 2: The leading monomial in the expansion of is . Proof: We have The last equality holds if and only if We shall now prove the theorem: 1. Let be a symmetric polynomial in variables . The proof is by strong induction on (see definition). If then is a constant polynomial, and it is easy to show the algorithm holds. Let us assume the algorithm holds for all symmetric polynomials with , for some . We will show that the algorithm holds also for a symmetric polynomial with , such that . By lemma 2, we get: The function is a polynomial, since . In addition, by the properties of symmetric polynomials is a symmetric polynomial in variables , therefore so is . The polynomials both contain , hence it is cancelled in their subtraction. If then . If then , meaning . Thus, the inductive assumption holds for , and therefore the algorithm yields a polynomial such that 2. The properties of the theorem hold: By definition, the degree of is and the degree of is at least . If has integer coefficients then is an integer. Therefore too has integer coefficients. Important results 0 95% #FFFFFF Theorem: Let be a field, and let be a polynomial of degree with roots . Let be a symmetric polynomial. Then . Proof. By Vieta's formulae, we get: By the fundamental theorem above, can be expressed as a polynomial 0 95% #FFFFFF Theorem: Let be a field, and let be a polynomial of degree with roots . Let , and let be the sums/products of every of the roots (namely ). Then there exists a monic polynomial of degree with roots . Proof. We will show that By Vieta's formulae, its coefficients are all symmetric polynomials in . Let be a symmetric polynomial, and let be the sums/products of every of the variables . Then can be expressed as a polynomial It is easy to see that by applying a permutation on , we also apply a permutation on . Hence is a symmetric polynomial, and by the previous theorem we get Fundamental Theorem of Symmetric Polynomials Elementary Symmetric Polynomials