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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 OEx 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 OEx 1 ; : : : ; x n Ł is a vector space over K and KOEx 1 ; : : : ; x n Ł is the direct sum over over d 0 of the subspaces K d OEx 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 .f /.x 1 ; : : : ; x n / D f .x .1/ ; : : : ; x .n/ / . For 2 S n , .x ˛ / D x ˛ 1 .1/ : : : x ˛ n .n/ . A polynomial or rational function is called symmetric if it is fixed by the S
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Unformatted text preview: S n action. The set of symmetric polynomials is denoted K S Œx 1 ;:::;x n Ł , and the set of symmetric rational functions is denoted K S .x 1 ;:::;x n / . Note that for each d , K d Œx 1 ;:::;x n Ł is invariant under the action of S n , and K S Œx 1 ;:::;x n Ł is the direct sum of the vector subspaces K S d Œx 1 ;:::;x n Ł D K d Œx 1 ;:::;x n Ł \ K S Œx 1 ;:::;x n Ł for d ² . See Exercise 9.6.3 . Lemma 9.6.1. (a) The action of S n on KŒx 1 ;:::;x n Ł is an action by ring automor-phisms; the action of of S n on K.x 1 ;:::;x n / is an action by by field automorphisms. (b) K S Œx 1 ;:::;x n Ł is a subring of KŒx 1 ;:::;x n Ł and K S .x 1 ;:::;x n / is a subfield of K.x 1 ;:::;x n / . (c) The field of symmetric rational functions is the field of fractions of the ring of symmetric polynomials in n-variables....
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