The Steenrod squares $Sq^i: H^n(-;\mathbb{F}_2) \to
H^{n+i}(-;\mathbb{F}_2)$ are fundamental cohomological
operations. By Yoneda lemma, it induces a map between the
Eilenberg-MacLane spaces $K(\mathbb{F}_2; n) \to
K(\mathbb{F}_2; n +i)$. By Dold-Kan correspondence, this map
should be expressible as a chain map (if I'm not mistaken): $$\widehat{Sq^i}: \mathbb{F}_2[-n]
\to \mathbb{F}_2[-(n+i)].$$

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**Question**: How to explicitly describe $\widehat{Sq^i}$? Is it much harder to do this for other finite fields $\mathbb{F}_q$?