He was a perfectionist toiling in an imperfect field.
2.
By definition, is perfect if and only if its separable and algebraic closures coincide ( in particular, the notion of a separable closure is only interesting for imperfect fields ).
3.
A similar construction works using a primitive nontrivial purely inseparable finite extension of an imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bit more complicated than in the preceding quadratic examples.
4.
Over imperfect fields of characteristic 2 and 3 there are some extra exotic pseudo-reductive groups coming from the existence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type F?in characteristic 2, and between groups of type G?in characteristic 3, using a construction analogous to that of the Ree groups.
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