  First, by inspiration of the results of Wood \cite{differential,problems},
but with the methods of non-commutative geometry and different approach, we
extend the coefficients of the Steenrod squaring operations from the filed
$\mathbb{F}_2$ to the dyadic integers $\mathbb{Z}_2$ and call the resulted
operations the dyadic Steenrod squares, denoted by $Jq^k$. The derivation-like
operations $Jq^k$ generate a graded algebra, called the dyadic Steenrod
algebra, denoted by $\mathcal{J}_2$ acting on the polynomials
$\mathbb{Z}_2[\xi_1, \dots, \xi_n]$. Being $\mathcal{J}_2$ an Ore domain,
enable us to localize $\mathcal{J}_2$ which leads to the appearance of the
integration-like operations $Jq^{-k}$ satisfying the
$Jq^{-k}Jq^k=1=Jq^kJq^{-k}$. These operations are enough to exhibit a kind of
differential equation, the dyadic Steenrod ordinary differential equation. Then
we prove that the completion of $\mathbb{Z}_2[\xi_1, \dots, \xi_n]$ in the
linear transformation norm coincides with a certain Tate algebra. Therefore,
the rigid analytic geometry is closely related to the dyadic Steenrod algebra.
Finally, we define the Adem norm $\| \ \|_A$ in which the completion of
$\mathbb{Z}_2[\xi_1, \dots, \xi_n]$ is
$\mathbb{Z}_2\llbracket\xi_1,\dots,\xi_n\rrbracket$, the $n$-variable formal
power series. We surprisingly prove that an element $f \in
\mathbb{Z}_2\llbracket \xi_1,\dots,\xi_n\rrbracket$ is hit if and only if
$\|f\|_A<1$. This suggests new techniques for the traditional Peterson hit
problem in finding the bases for the cohit modules.
