Galois theory

A. R. Magid, The Separable Galois Theory of Commutative Rings (1974)... 

Since separable algebras over k all look basically the same over ks, classifying them is a descent problem of the type we will study more generally in Chapter 17. But since usual Galois theory already classifies separable fields,... 

If char(k) = 0, all finite group schemes are in fact etale (11.4), and the other types do not occur. When char(lc) = p, however, we know examples of all four types Z/qZ with q prime to p is etale with etale dual i, while Z/pZ is etale with connected dual pp and vice versa, and p = a is connected with connected dual. The Galois theory of (6.4) describes the first two types, and also (after dualizing) the third. The fourth requires a theory of its own the groups are classified by modules over a certain ring, Dieudonne modules. ... 

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

The assumption that G is connected can be dropped. Indeed, let F° be the field fixed by G°. The finite group G/G° is solvable, so by ordinary Galois theory we can get F° from F by adjoining various n-th roots. These can all be constructed by uiln — exp J (u /nu), and then the preceding argument takes us on from F° to L. 

The essentials of field theory (Galois theory, separability, transcendence degree). 

Proof Note that Rev Et(S)= Rev Et(T). The first assertion follows from general Galois theory (SGA 1 V, 6). 

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