Ramified extensions of degree $p$ and their Hopf–Galois module structure

Elder, G. Griffith

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 19-40.

Les extensions cycliques ramifiées $L/K$ de degré $p$ d’un corps local dont la caractéristique résiduelle est $p$ sont plutôt bien comprises. Elles sont définies par une équation d’Artin–Schreier sauf lorsque $char\left(K\right)=0$ et $L=K\left(\sqrt[p]{{\pi }_K}\right)$ pour une certaine uniformisante ${\pi }_K\in K$. De plus, depuis les travaux de Bertrandias–Ferton ($char\left(K\right)=0$) puis Aiba ($char\left(K\right)=p$), plusieurs résultats sont connus sur la structure galoisienne des idéaux de telles extensions : on sait par exemple décrire la structure de chaque idéal ${𝔓}_L^n$ comme module sur son ordre associé ${𝔄}_{K\left[G\right]}\left(n\right)=\{x\in K\left[G\right]:x{𝔓}_L^n\subseteq {𝔓}_L^n\}$, où $G=Gal(L/K)$. Le but de cet article est d’étendre ces résultats aux extensions séparables et ramifiées de degré $p$ qui ne sont pas galoisiennes.

Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. They are defined by an Artin–Schreier equation, unless $char\left(K\right)=0$ and $L=K\left(\sqrt[p]{{\pi }_K}\right)$ for some prime element ${\pi }_K\in K$. Moreover, through the work of Bertrandias–Ferton ($char\left(K\right)=0$) and Aiba ($char\left(K\right)=p$), much is known about the Galois module structure of the ideals in such extensions: the structure of each ideal ${𝔓}_L^n$ as a module over its associated order ${𝔄}_{K\left[G\right]}\left(n\right)=\{x\in K\left[G\right]:x{𝔓}_L^n\subseteq {𝔓}_L^n\}$ where $G=Gal(L/K)$. The purpose of this paper is to extend these results to separable, ramified extensions of degree $p$ that are not Galois.

Classification : 11S15, 11R33, 16T05
Mots clés : Artin–Schreier equation, Galois module structure

[1] Aiba, Akira Artin-Schreier extensions and Galois module structure, J. Number Theory, Volume 102 (2003) no. 1, pp. 118-124 | DOI | MR | Zbl

[2] Amano, Shigeru Eisenstein equations of degree $p$ in a $𝔭$-adic field, J. Fac. Sci. Univ. Tokyo, Sect. I A, Volume 18 (1971), pp. 1-21 | MR | Zbl

[3] Artin, Emil; Tate, John Class field theory, Advanced Book Classics, Addison-Wesley Publishing Company, 1990, xxxviii+259 pages | Zbl

[4] Bertrandias, Françoise; Bertrandias, Jean-Paul; Ferton, Marie-Josée Sur l’anneau des entiers d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci., Paris, Sér. A, Volume 274 (1972), pp. 1388-1391 | Zbl

[5] Byott, Nigel P. Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications, J. Théor. Nombres Bordx., Volume 9 (1997) no. 1, pp. 201-219 | DOI | MR | Zbl

[6] Byott, Nigel P.; Childs, Lindsay N.; Elder, G. Griffith Scaffolds and generalized integral Galois modules structure (2017) (to appear in Ann. Inst. Fourier, https://arxiv.org/abs/1308.2088v3) | Numdam | Zbl

[7] Childs, Lindsay N. On the Hopf Galois theory for separable field extensions, Commun. Algebra, Volume 17 (1989) no. 4, pp. 809-825 | DOI | MR | Zbl

[8] Childs, Lindsay N. Taming wild extensions: Hopf algebras and local Galois module theory, Mathematical Surveys and Monographs, 80, American Mathematical Society, 2000, viii+215 pages | MR | Zbl

[9] Dummit, David S.; Foote, Richard M. Abstract algebra, Wiley International Edition, 2004, xii+932 pages | Zbl

[10] Elder, G. Griffith Galois scaffolding in one-dimensional elementary abelian extensions, Proc. Am. Math. Soc., Volume 137 (2009) no. 4, pp. 1193-1203 | DOI | MR | Zbl

[11] Ferton, Marie-Josée Sur les idéaux d’une extension cyclique de degré premier d’un corps local, C. R. Acad. Sci., Paris, Sér. A, Volume 276 (1973), pp. 1483-1486 | MR | Zbl

[12] Fesenko, Ivan Borisovich; Vostokov, Sergei Vladimirovich Local fields and their extensions, Translations of Mathematical Monograph, 121, Translations of Mathematical Monograph, 2002, xi+345 pages | MR | Zbl

[13] Greither, Cornelius; Pareigis, Bodo Hopf Galois theory for separable field extensions, J. Algebra, Volume 106 (1987) no. 1, pp. 239-258 | DOI | MR | Zbl

[14] Helou, Charles On the ramification breaks, Commun. Algebra, Volume 19 (1991) no. 8, pp. 2267-2279 | DOI | MR | Zbl

[15] Huynh, Duc Van Artin-Schreier extensions and generalized associated orders, J. Number Theory, Volume 136 (2014), pp. 28-45 | DOI | MR | Zbl

[16] Jones, John W.; Roberts, David P. A database of local fields, J. Symb. Comput., Volume 41 (2006) no. 1, pp. 80-97 | DOI | MR | Zbl

[17] Lang, Serge Algebra, Graduate Texts in Mathematics, 211, Springer, 2002, xv+914 pages | Zbl

[18] Lubin, Jonathan The local Kronecker-Weber theorem, Trans. Am. Math. Soc., Volume 267 (1981), pp. 133-138 | DOI | MR | Zbl

[19] Lubin, Jonathan Elementary analytic methods in higher ramification theory, J. Number Theory, Volume 133 (2013) no. 3, pp. 983-999 | DOI | MR | Zbl

[20] Luca, Florian; Rzedowski-Calderón, Martha; Maldonado-Ramírez, Myriam Rosalía A generalization of a lemma of Sullivan, Commun. Algebra, Volume 40 (2012) no. 7, pp. 2301-2308 | DOI | MR | Zbl

[21] MacKenzie, Robert E.; Whaples, George Artin-Scheier equations in characteristic zero, Am. J. Math., Volume 78 (1956), pp. 473-485 | DOI | Zbl

[22] Marklove, Maria Local Galois module structure in characteristic $p$, University of Exeter (UK) (2013) (Ph. D. Thesis) | MR

[23] Serre, Jean-Pierre Local fields., Graduate Texts in Mathematics, 67, Springer, 1979, viii+241 pages | Zbl

[24] de Smit, Bart; Thomas, Lara Local Galois module structure in positive characteristic and continued fractions, Arch. Math., Volume 88 (2007) no. 3, pp. 207-219 | DOI | MR | Zbl

[25] Sullivan, Francis J. $p$-torsion in the class group of curves with too many automorphisms, Arch. Math., Volume 26 (1975), pp. 253-261 | DOI | MR | Zbl

[26] The LMFDB Collaboration The L-functions and Modular Forms Database, 2016 (http://www.lmfdb.org)