  In the spirit of the work of Grothendieck, we introduce and study natural
symmetric n-fold tensor norms. We prove that there are exactly six natural
symmetric tensor norms for $n\ge 3$, a noteworthy difference with the 2-fold
case in which there are four. We also describe the polynomial ideals associated
to these natural symmetric tensor norms. Using a symmetric version of a result
of Carne, we establish which natural symmetric tensor norms preserve Banach
algebras.
