Calculations with Symmetric Functions
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Determine whether w1,...,wn generate the ring of symmetric functions where w_k=u_1^k+⋯+u_n^k
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Solution Summary
This solution helps calculate symmetric functions.
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We assume that all polynomials are considered over a field F.
The answer to the posed question depends on whether the characteristic of F is zero.
Let the characteristic of F be zero. Let us denote by R the subring of F[u_1,...,u_n] generated by w_1,...,w_n .
According to the Newton's formulas, we have
ke_k (u_1,...,u_n)= ∑_(i=1)^k▒〖(-1)〗^(i-1) e_(k-i)(u_1,...,u_n)w_i(u_1,...,u_n), (1)
for any k≥1. Here e_k (u_1,...,u_n) denotes the k-th elementary symmetric ...
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