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

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