Symmetric algebra

Summary

The symmetric algebra of a vector space $V$ is the tensor algebra of $V$ made commutative. It is the most general commutative algebra generated by $V$. If $V$ has a finite basis $x_1, \ldots, x_n$, it is the polynomial ring $F[x_1, \ldots, x_n]$, so it gives a coordinate-free description of polynomials.