Serre polynomial generators for mod-two Eilenberg–MacLane cohomology (${\mathbb{F}}_2[{\mathrm{Sq}}^I{u}_n:e(I)<n]$)

For $n\ge 1$, the mod-two cohomology of $K(\mathbb{Z}/2,n)$ is polynomial on the admissible operations in the displayed strict-excess range, including the empty operation. It is obtained from path-space transgressions and compatibility with Steenrod squares. Products must still be included when listing low-degree vector-space bases.

For $n\ge 2$, the positive-degree dimensions in degrees $n,n+1,n+2,n+3$ are $1,1,1,2$. The first three classes are $u,{\mathrm{Sq}}^{1}u,{\mathrm{Sq}}^{2}u$. In degree $n+3$, use ${\mathrm{Sq}}^{3}u$ and ${\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ for $n\ge 3$; at $n=2$ replace the unstable first class by $u{\mathrm{Sq}}^{1}u$. At $n=2$ the degree-four class is $u^2$, and at $n=3$ the degree-six class ${\mathrm{Sq}}^{3}u$ is $u^2$.