Integral Galois module structure for elementary abelian extensions with a Galois scaffold

Byott, Nigel; Elder, G.

doi:10.1090/S0002-9939-2014-12126-5

Integral Galois module structure for elementary abelian extensions with a Galois scaffold
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by Nigel P. Byott and G. Griffith Elder PDF

Proc. Amer. Math. Soc. 142 (2014), 3705-3712 Request permission

Abstract:

This paper justifies an assertion by [Elder, Proc. Amer. Math. Soc., 2009] that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in loc. cit., we give a necessary and sufficient condition for the valuation ring $\mathfrak {O}_L$ to be free over its associated order $\mathfrak {A}_{L/K}$ in $K[\mathrm {Gal}(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata concerning a class of cyclic Kummer extensions in characteristic zero.

References

Akira Aiba, Artin-Schreier extensions and Galois module structure, J. Number Theory 102 (2003), no. 1, 118–124. MR 1994476, DOI 10.1016/S0022-314X(03)00083-0
Françoise Bertrandias, Jean-Paul Bertrandias, and Marie-Josée Ferton, Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1388–A1391 (French). MR 296048
Françoise Bertrandias and Marie-Josée Ferton, Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A1330–A1333 (French). MR 296047
Nigel P. Byott, On the integral Galois module structure of cyclic extensions of $p$-adic fields, Q. J. Math. 59 (2008), no. 2, 149–162. MR 2428073, DOI 10.1093/qmath/ham037
Nigel P. Byott and G. Griffith Elder, Galois scaffolds and Galois module structure in extensions of characteristic $p$ local fields of degree $p^2$, J. Number Theory 133 (2013), no. 11, 3598–3610. MR 3084290, DOI 10.1016/j.jnt.2013.04.021
Bart de Smit and Lara Thomas, Local Galois module structure in positive characteristic and continued fractions, Arch. Math. (Basel) 88 (2007), no. 3, 207–219. MR 2305599, DOI 10.1007/s00013-006-1939-8
G. Griffith Elder, Galois scaffolding in one-dimensional elementary abelian extensions, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1193–1203. MR 2465640, DOI 10.1090/S0002-9939-08-09710-4
Bruno Martel, Sur l’anneau des entiers d’une extension biquadratique d’un corps $2$-adique, C. R. Acad. Sci. Paris Sér. A 278 (1974), 117–120 (French). MR 337906
Yoshimasa Miyata, On the Galois module structure of ideals and rings of all integers of $\mathfrak {p}$-adic number fields, J. Algebra 177 (1995), no. 3, 627–646. MR 1358477, DOI 10.1006/jabr.1995.1320
Y. Miyata, On the module structure of rings of integers in $\mathfrak {p}$-adic number fields over associated orders, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 2, 199–212. MR 1490195, DOI 10.1017/S0305004197002016
Yoshimasa Miyata, Maximal tame extensions over Hopf orders in rings of integers of ${\mathfrak {p}}$-adic number fields, J. Algebra 276 (2004), no. 2, 794–825. MR 2058468, DOI 10.1016/j.jalgebra.2003.10.013
Paulo Ribenboim, The book of prime number records, 2nd ed., Springer-Verlag, New York, 1989. MR 1016815, DOI 10.1007/978-1-4684-0507-1