Hopf orders, applications of

Let be a finite extension of the -adic rationals endowed with the -adic valuation with and let be its ring of integers (cf. also Extension of a field; Norm on a field; -adic number). Let be the ramification index of in and assume that contains a primitive rd root of unity. If is an -Hopf order in (cf. Hopf order), then the group scheme can be resolved, i.e., involved in a short exact sequence of group schemes (taken in the flat topology). Here, and are represented by the -Hopf algebras and , respectively ( an indeterminate). One uses this short exact sequence in the long exact sequence in cohomology to construct , which is identified with the group of -Galois extensions of (cf. also Galois extension). One has , where the class corresponds to the isomorphism class of the -Galois extension

with (see [a2]). Moreover, if , , is the dual of the Larson order (cf. Hopf order), then can be involved in the short exact sequence of group schemes , where and are represented by the -Hopf algebras and for appropriate polynomials , respectively (see [a3]). Work has been completed to obtain a resolution of when is an arbitrary -Hopf order in (see [a4]). This yields (via the long exact sequence in cohomology) a characterization of all -Galois extensions. In general, if is an -Hopf order in , then is an -Galois extension if and only if is an -Galois algebra (see [a1]).

Examples of -Galois extensions where is a Tate–Oort–Larson order in can be recovered using the corresponding classification theorem. For example, if is a Kummer extension of prime degree, then is an -Galois algebra if and only if the ramification number of satisfies (see [a1]). Therefore, is an -Galois extension, thus

for some parameter . Knowledge of Tate–Oort Galois extensions can also be used to characterize the ring of integers of certain degree- extensions. For example, if , , is a Greither order with and , then there exists a finite extension so that is an -Galois algebra. In this case, is of the form

where is an element in and is an -Galois extension (see [a2]).

References

[a1]	L. Childs, "Taming wild extensions with Hopf algebras" Trans. Amer. Math. Soc. , 304 (1987)
[a2]	C. Greither, "Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring" Math. Z. , 210 (1992) pp. 37–67
[a3]	T. Sekiguchi, N. Suwa, "Théories de Kummer–Artin–Schreier–Witt" C.R. Acad. Sci. Ser. I , 319 (1994) pp. 1–21
[a4]	R.G. Underwood, "The group of Galois extensions in " Trans. Amer. Math. Soc. , 349 (1997) pp. 1503–1514