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Unformatted text preview: Theorem 9.6.6. The set of elementary symmetric functions f 1 ;:::; n g in Kx 1 ;:::;x n is algebraically independent over K , and gener-ates K S x 1 ;:::;x n as a ring. Consequently, K. 1 ;:::; n / D K S .x 1 ;:::;x n / . The algebraic independence of the i is the same as linear indepen-dence of the monic monomials in the i . First, we establish an indexing system for the monic monomials: A partition is a nite decreasing se-quence of nonnegative integers, D . 1 ;:::; k / . We can picture a parti-tion by means of an M-by-N matrix of zeroes and ones, where M k...
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- Fall '08