Symmetric functions, $m$-sets, and Galois groups
David Casperson, John McKay · Mathematics of Computation · 1994
Given the elementary symmetric functions in $\{ {r_i}\} \;(i = 1, \ldots ,n)$, we describe algorithms to compute the elementary symmetric functions in the products $\{ {r_{{i_1}}}{r_{{i_2}}} \cdots {r_{{i_m}}}\} \;(1 \leq {i_1} < \cdots < {i_m} \leq n)$ and in the sums $\{ {r_{{i_1}}} + {r_{{i_2}}} + \cdots + {r_{{i_m}}}\} \;(1 \leq {i_1} < \cdots < {i_m} \leq n)$. The computation is performed over the coefficient ring generated by the elementary symmetric functions. We apply FFT multiplication of series to reduce the complexity of the algorithm for sums. An application to computing Galois groups is given.