  In this work, we study the classification of differential graded algebras
over $\mathbb{Z}$ (DGAs) whose homology is $\mathbb{F}_p[x]$, i.e. the
polynomial algebra over $\mathbb{F}_p$ on a single generator. This
classification problem was left open in work of Dwyer, Greenlees and Iyengar.
  For $\lvert y_{2p-2} \rvert = 2p-2$, we show that there is a unique
non-formal DGA with homology $\mathbb{F}_p[y_{2p-2}]$ and a non-formal $2p-2$
Postnikov section. Among a classification result, this provides the first
example of a non-formal DGA with homology $\mathbb{F}_p[x]$. By duality, this
also shows that there is a non-formal DGA whose homology is an exterior algebra
over $\mathbb{F}_p$ with a generator in degree $-(2p-1)$.
  Considering the classification of the ring spectra corresponding to these
DGAs, we show that every $E_2$ DGA with homology $\mathbb{F}_p[x]$ (with no
restrictions on $\lvert x \rvert$) is topologically equivalent to the formal
DGA with homology $\mathbb{F}_p[x]$, i.e. they are topologically formal. This
follows by a theorem of Hopkins and Mahowald.

Algebraic Topology