Is polynomial algebra the only minimal dense subalgebra of $C^{\infty}(U)$ Let $U$ be connected open set of $\mathbb{R}^{n}$. Consider $C^{\infty}(U)$ with sup norm, we said $A\subset C^{\infty}(U)$ is a minimal dense subalgebra of $C^{\infty}(U)$ if and only if any subalgebra contain in $A$ is not dense in $C^{\infty}(U)$. I want to ask are the polynomial algebra of $U$, either with coefficients in $\mathbb{Q}$ or $\mathbb{R}$\ $\mathbb{Q}$, the only two minimal dense subalgebra of $C^{\infty}(U)$?

For proper U, polynomials in even powers or trigonometric polynomials i.e. span of ${e^{ikt}}$ will do. Also, if $U=[a,b]\subset (0,1)$ then polynomials with integer coefficients are good.
