  Exponential sums have applications to a variety of scientific fields,
including, but not limited to, cryptography, coding theory and information
theory. Closed formulas for exponential sums of symmetric Boolean functions
were found by Cai, Green and Thierauf in the late 1990's. Their closed formulas
imply that these exponential sums are linear recursive. The linear recursivity
of these sums has been exploited in numerous papers and has been used to
compute the asymptotic behavior of such sequences. In this article, we extend
the result of Cai, Green and Thierauf, that is, we find closed formulas for
exponential sums of symmetric polynomials over any Galois fields. Our result
also implies that the recursive nature of these sequences is not unique to the
binary field, as they are also linear recursive over any finite field. In fact,
we provide explicit linear recurrences with integer coefficients for such
sequences. As a byproduct of our results, we discover a link between
exponential sums of symmetric polynomials over Galois fields and a problem for
multinomial coefficients which similar to the problem of bisecting binomial
coefficients.
