Complex group algebra of $S_3$ isomorphic to $\mathbb{C}[x]/(p)$?

Question

For the symmetric group on two letters, $S_2$, there exists an isomorphism from the complex group algebra $\mathbb{C}[S_2]$ to the complex polynomial algebra $\mathbb{C}[x]/(x^2 - 2x)$ by taking $e + (1,2)$ in $\mathbb{C}[S_2]$ to $x$ in $\mathbb{C}[x]/(x^2 - 2x)$. Does there exist a polynomial $p$ such that $\mathbb{C}[S_3] \cong \mathbb{C}[x]/(p)$?

Answer 1

$S_3$ is non-Abelian, so that $\Bbb C[S_3]$ is non-commutative.

$\Bbb C[X]$ is commutative, so that $\Bbb C[X]/(p)$ is commutative too.