SYMMETRICAL FUNCTIONS, M-SETS, AND GALOIS-GROUPS

Given the elementary symmetric functions in {r(i)} (i = 1, ..., n), we describe algorithms to compute the elementary symmetric functions in the products {r(i1), r(i2), ... r(im)} (1 less than or equal to i(i) < ... < i(m) less than or equal to (n)) and in the sums {r(i1) + r(i2) + ... + r(im)} (1 less than or equal to i(1) < ... < i(m) less than or equal to n). The computation is performed over the coefficient ring g enerated by the elementary symmetric functions. We apply FFT multiplic ation of series to reduce the complexity of the algorithm for sums. An application to computing Galois groups is given.