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For a finite cyclic Galois extension of fields $$K/k$$K/k of degree $$n$$n and a separable polynomial of degree $$dn$$dn or $$dn - 1$$dn-1, we construct an explicit smooth compactification $$X \rightarrow {\mathbb P}^1_k$$X→Pk1 of the affine normic bundle $$X_0$$X0 given by $$\begin{aligned} {{\mathrm{N}}}_{K/k}(\mathbf {z}) = P(x) \ne 0, \end{aligned}$$NK/k(z)=P(x)≠0,extending the map $$X_0 \rightarrow {\mathbb A}_k^1$$X0→Ak1, where $$(\mathbf {z},x) \mapsto x$$(z,x)↦x. The construction makes no assumption of the characteristic of $$k$$k, making it a suitable departure point for studying the arithmetic of smooth compactifications of $$X_0$$X0 over global fields of positive characteristic.

Smooth compactifications of certain normic bundles