Remark on the Serre \((\text{mod }5)\)-invariant for groups of type \(E_ 8\)
Let \(G_0\) be a linear simple algebraic group of type \(E_8\) defined and factorable over an arbitrary field \(F\) of characteristic \(\neq 2, 3, 5\), and let \(\mu_5 \subset F^s\) be a \(\text{Gal} (F^s/F)\)- module of roots of unity of power 5. (Here we denote by \(F^s\) the separable closure of the field \(F\).) For arbitrary extensions \(L\) of the field \(F\), Serre and Rost constructed mappings \(\varphi_L : H^1 (L, G_0) \to H^3 (L, \mu^{\otimes 2}_5)\) that take a unity element to a unity element and have natural functor properties. The purpose of the present paper is to describe the kernel of the mapping \(\varphi = \varphi_F\).
scientific article; zbMATH DE number 830695