Artin-Schreier extensions and thier applications

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A Galois extension E/F of fields is called a cyclic extension if the Galois group is cyclic. Assume that p > 0 is the characteristic of our fields and n is the degree of the field extension E/F. If n is relatively prime to p, and there is a primitive n th root of unity in F, then E/F is a Kummer extension, i.e. E = F(y) with y n ∈ F. If n = p, then E/F is an Artin-Schreier extension, i.e. E = F(y) with y p y ∈ F. Finally, if n = p a for a > 1, then the extension E/F can be described in terms of Witt vectors. For these facts, see [34, Section VI.7].