2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146833A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gröbner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$.preliminary abstract, 10 pagesCommutative AlgebraCombinatorics13E05, 13E15, 20B30, 06A07text