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Generalization of the $${\varvec{lq}}$$lq-modular closure theorem and applications
- Source :
- Archiv der Mathematik. 112:361-370
- Publication Year :
- 2018
- Publisher :
- Springer Science and Business Media LLC, 2018.
-
Abstract
- Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.
- Subjects :
- Discrete mathematics
Modularity (networks)
business.industry
Generalization
General Mathematics
010102 general mathematics
Separable extension
Field (mathematics)
Extension (predicate logic)
Modular design
01 natural sciences
Integer
Field extension
0103 physical sciences
010307 mathematical physics
0101 mathematics
business
Mathematics
Subjects
Details
- ISSN :
- 14208938 and 0003889X
- Volume :
- 112
- Database :
- OpenAIRE
- Journal :
- Archiv der Mathematik
- Accession number :
- edsair.doi...........597285b24468be3f77d8d4ea5cebd104