College Algebra Exam Review 430

College Algebra Exam Review 430 - S n action. The set of...

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440 9. FIELD EXTENSIONS – SECOND LOOK (b) Show that ± 7! ± j B is an injective homomorphism of Aut A .A ± B/ into Aut A \ B .B/ . (c) Check surjectivity of ± 7! ± j B as follows: Let G 0 D f ± j B W ± 2 Aut A .A ± B/ g : Then Fix .G 0 / D Fix . Aut A .A ± B// \ B D A \ B: Therefore, by the Galois correspondence, G 0 D Aut A \ B .B/ . 9.6. Symmetric Functions Let K be any field, and let x 1 ;:::;x n be variables. For a vector ˛ D 1 ;:::;˛ n / , with nonnegative integer entries, let x ˛ D x ˛ 1 1 :::x ˛ n n . The total degree of the monic monomial x ˛ is j ˛ j D P ˛ i . A polynomial is said to be homogeneous of total degree d if it is a linear combination of monomials x ˛ of total degree d . Write K d Œx 1 ;:::;x n Ł for the set of polynomials in n variables that are homogeneous of total degree d or identically zero. Then K d Œx 1 ;:::;x n Ł is a vector space over K and KŒx 1 ;:::;x n Ł is the direct sum over over d ² 0 of the subspaces K d Œx 1 ;:::;x n Ł ; see Exercise 9.6.1 . The symmetric group S n acts on polynomials and rational functions in n variables over K by ².f /.x 1 ;:::;x n / D f.x ±.1/ ;:::;x ±.n/ / . For ² 2 S n , ².x ˛ / D x ˛ 1 ±.1/ :::x ˛ n ±.n/ . A polynomial or rational function is called symmetric if it is fixed by the
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Unformatted text preview: S n action. The set of symmetric polynomials is denoted K S x 1 ;:::;x n , and the set of symmetric rational functions is denoted K S .x 1 ;:::;x n / . Note that for each d , K d x 1 ;:::;x n is invariant under the action of S n , and K S x 1 ;:::;x n is the direct sum of the vector subspaces K S d x 1 ;:::;x n D K d x 1 ;:::;x n \ K S x 1 ;:::;x n for d . See Exercise 9.6.3 . Lemma 9.6.1. (a) The action of S n on Kx 1 ;:::;x n is an action by ring automor-phisms; the action of of S n on K.x 1 ;:::;x n / is an action by by eld automorphisms. (b) K S x 1 ;:::;x n is a subring of Kx 1 ;:::;x n and K S .x 1 ;:::;x n / is a subeld of K.x 1 ;:::;x n / . (c) The eld of symmetric rational functions is the eld of fractions of the ring of symmetric polynomials in n-variables....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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