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 $C^{\infty}(U)$?

In orded for the sup norm  to be well defined for polinomials one needs to assume that $U$ is bounded.  On the other hand the connectedness of $U$ seems irrelevant here.
In that case, the algebra of polynomials is dense in the sup norm by Stone-Weierstrass' Theorem, but it is not minimal.  The reason is that, for every odd number $k$, the set of polynomials of the form
$$
  p(x) = \sum_{n=0}^d a_n x^{kn},
  $$
is dense (again by Stone-Weierstrass) and it is a proper subalgebra of the polinomial algebra (this works also if $k$ is even and $U\subseteq (0,+\infty)$).
