A generalization of Kummer theory to Hopf-Galois extensions

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10489835" target="_blank" >RIV/00216208:11320/24:10489835 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=E17xcV4ciL" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=E17xcV4ciL</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jalgebra.2024.07.017" target="_blank" >10.1016/j.jalgebra.2024.07.017</a>

Jazyk výsledku

angličtina

Název v původním jazyce

A generalization of Kummer theory to Hopf-Galois extensions

Popis výsledku v původním jazyce

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an H-Galois extension L/K is H-Kummer if L can be generated by adjoining to K a finite set S of eigenvectors for the action of the Hopf algebra Hon L. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of H-Kummer extensions L/K as radical extensions that are linearly disjoint with the n-th cyclotomic extension of K. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the n-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions L-1/K, L-2/K such that L-1 boolean AND M-2 = L-2 boolean AND M-1 = K, where Mi is a field such that LiMi = (L) over tilde (i), the normal closure of L-i/K. When L/K is an extension of number or p-adic fields, we shall derive criteria on the freeness of the ring of integers O-L over its associated order in an almost classically Galois structure on L/K. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Název v anglickém jazyce

A generalization of Kummer theory to Hopf-Galois extensions

Popis výsledku anglicky

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an H-Galois extension L/K is H-Kummer if L can be generated by adjoining to K a finite set S of eigenvectors for the action of the Hopf algebra Hon L. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of H-Kummer extensions L/K as radical extensions that are linearly disjoint with the n-th cyclotomic extension of K. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the n-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions L-1/K, L-2/K such that L-1 boolean AND M-2 = L-2 boolean AND M-1 = K, where Mi is a field such that LiMi = (L) over tilde (i), the normal closure of L-i/K. When L/K is an extension of number or p-adic fields, we shall derive criteria on the freeness of the ring of integers O-L over its associated order in an almost classically Galois structure on L/K. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GM21-00420M" target="_blank" >GM21-00420M: Univerzální kvadratické formy a třídová čísla</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Algebra

ISSN

0021-8693

e-ISSN

1090-266X

Svazek periodika

660

Číslo periodika v rámci svazku

December 2024

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

46

Strana od-do

190-235

Kód UT WoS článku

001285867600001

EID výsledku v databázi Scopus

2-s2.0-85199959732