Normalized Derivations for Milnor's Primitive Operations on the Dickson Algebra and Applications

We study the action of the Steenrod–Milnor operation ${\mathrm{S}\mathrm{t}}^{\mathrm{\varnothing },{\mathrm{\Delta }}_i}$ on the Dickson algebra $D_n$ over ${\mathbb{F}}_p$. Our main observation is that normalizing by the Dickson invariant $Q_{n,0}$ yields a genuine derivation on the localization $D_n[Q_{n,0}^{-1}]$. This viewpoint provides a transparent framework to derive a closed formula for all higher iterates of ${\mathrm{S}\mathrm{t}}^{\mathrm{\varnothing },{\mathrm{\Delta }}_i}$ on the Dickson generators. Consequently, we establish the vanishing condition $({\mathrm{S}\mathrm{t}}^{\mathrm{\varnothing },{\mathrm{\Delta }}_i}{)}^m=0$ on the generators for $m\ge p$, and the stronger global operator identity $({\mathrm{S}\mathrm{t}}^{\mathrm{\varnothing },{\mathrm{\Delta }}_i}{)}^p=0$ on all of $D_n$. Furthermore, upon localizing by $R_{n,i}^p$, the normalized action becomes Euler-type. This allows us to exactly determine the kernel and image of the derivation in the classical range $2\le i<n$, and describe them via an auxiliary grading when $i=n$. As an application, our general formalism recovers several known first-order formulas and upgrades them to closed expressions for all higher iterates. Finally, we present an ordinary Koszul-type construction attached to normalized-ratio coefficients, providing a structural analogy to Margolis homology for operations on the $\xi$-side that do not necessarily square to zero.