no. 1
Galois scaffolds for $p$-extensions in characteristic $p$
Elder, G. Griffith1; Keating, Kevin2
1Department of Mathematics, University of Nebraska Omaha, Omaha, NE 68182 (USA)
2Department of Mathematics, University of Florida, Gainesville, FL 32611 (USA)
Annales de l'Institut Fourier, Volume 76 (2026) no. 1, pp. 397-423

Let $K$ be a local field of characteristic $p>0$ with perfect residue field and let $G$ be a finite $p$-group. In this paper we use Saltman’s construction of a generic $G$-extension of rings of characteristic $p$ to construct totally ramified $G$-extensions $L/K$ that have Galois scaffolds. We specialize this construction to produce $G$-extensions $L/K$ such that the ring of integers $\mathcal{O}_L$ is free of rank $1$ over its associated order $\mathcal{A}_0$, and extensions such that $\mathcal{A}_0$ is a Hopf order in the group ring $K[G]$.

Soit $K$ un corps local de caractéristique $p>0$ de corps résiduel parfait et soit $G$ un $p$-groupe fini. Dans cet article nous utilisons la construction de Saltman d’une $G$-extension générique d’anneaux de caractéristique $p$ pour construire des $G$-extensions $L/K$ totalement ramifiées qui ont un échafaudage galoisien. Nous spécialisons cette construction pour produire des $G$-extensions $L/K$ telles que l’anneau d’entiers $\mathcal{O}_L$ soit libre de rang $1$ sur son ordre associé $\mathcal{A}_0$, et des extensions telles que $\mathcal{A}_0$ soit un ordre de Hopf dans l’anneau de groupe $K[G]$.

Received:
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Accepted:
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Published online:
DOI: 10.5802/aif.3712
Classification: 11S15, 11R33, 14L15, 16T05, 11S23
Keywords: generic extensions, ramification, Galois module structure, Galois scaffold, Hopf order
Mots-clés : extension générique, ramification, structure du module galoisien, échafaudage galoisien, ordre de Hopf

Elder, G. Griffith  1 ; Keating, Kevin  2

1 Department of Mathematics, University of Nebraska Omaha, Omaha, NE 68182 (USA)
2 Department of Mathematics, University of Florida, Gainesville, FL 32611 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Elder, G. Griffith; Keating, Kevin. Galois scaffolds for $p$-extensions in characteristic $p$. Annales de l'Institut Fourier, Volume 76 (2026) no. 1, pp. 397-423. doi: 10.5802/aif.3712

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