Let $k$ be a field. We compute the set ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}$ ofnaivehomotopy classes of pointed $k$-scheme endomorphisms of the projective line ${𝐏}^1$. Our result compares well with Morel’s computation in [11] of thegroup${\left[{𝐏}^1,{𝐏}^1\right]}^{{𝐀}^1}$ of ${𝐀}^1$-homotopy classes of pointed endomorphisms of ${𝐏}^1$: the set ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}$ admits an a priori monoid structure such that the canonical map ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}\to {\left[{𝐏}^1,{𝐏}^1\right]}^{{𝐀}^1}$ is a group completion.

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group $\mathbf{G}$ over $\mathbf{Q}$ typically have splitting field with Galois group isomorphic to the Weyl group of $\mathbf{G}$, we construct an explicit monic integral polynomial of degree $240$ whose splitting field has Galois group the Weyl group of the exceptional group of type ${\mathbf{E}}_8$.