Counting Galois ${\mathbb{U}}_4({\mathbb{F}}_p)$-extensions using Massey products

We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$consistingofunipotentfourbyfourmatricesover${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb{U}}_4({\mathbb{F}}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condition when the $n$-fold Massey product of one-dimensional cohomological elements of $G$ with coefficients in ${\mathbb{F}}_p$, is defined. As an easy consequence, we determine those ${\mathbb{U}}_n({\mathbb{F}}_p)$ which occur as an epimorphic image of any given Demushkin group.