Reduced points of $\mathbb{E}_{\infty}$-rings in positive characteristic

We investigate whether an arbitrary non-zero $\mathbb{E}_\infty$-ring $A$ admits a reduced point, meaning an $\mathbb{E}_\infty$-map $A\to T$ such that $\pi_{\ast}T$ is a graded field. We show that if $2\in \pi_0A$ is not invertible, then $A$ admits a reduced point and as an application deduce that a free $A$-module on $n$ generators cannot be built from $n-1$ many cells. Perhaps surprisingly, the existence of reduced points completely fails at odd primes. More precisely, for any prime $p>2$, we construct a non-zero $\mathbb{E}_\infty$-ring over $\mathbb{F}_p$ which admits no map to an $\mathbb{E}_2$-algebra $T$ such that $\pi_0T$ is a field.