PAC fields over number fields
Moshe Jarden (2006)
Journal de Théorie des Nombres de Bordeaux
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We prove that if is a number field and is a Galois extension of which is not algebraically closed, then is not PAC over .
Moshe Jarden (2006)
Journal de Théorie des Nombres de Bordeaux
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We prove that if is a number field and is a Galois extension of which is not algebraically closed, then is not PAC over .
Claude Mitschi, Michael F. Singer (2002)
Annales de la Faculté des sciences de Toulouse : Mathématiques
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Nigel P. Byott (2011)
Journal de Théorie des Nombres de Bordeaux
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Let be a finite extension of discrete valuation rings of characteristic , and suppose that the corresponding extension of fields of fractions is separable and is -Galois for some -Hopf algebra . Let be the different of . We show that if is totally ramified and its degree is a power of , then any element of with generates as an -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois...
Martin Epkenhans (1997)
Acta Arithmetica
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Let mℤd ≀ mℤd ≀ mℤd ≀ m
Pierre Dèbes (1999)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Manjul Bhargava, Matthew Satriano (2014)
Journal of the European Mathematical Society
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We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an degree extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.
Núria Vila (1992)
Publicacions Matemàtiques
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The problem of the construction of number fields with Galois group over Q a given finite groups has made considerable progress in the recent years. The aim of this paper is to survey the current state of this problem, giving the most significant methods developed in connection with it.
Teresa Crespo, Zbigniew Hajto (2002)
Annales de l’institut Fourier
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An effective construction of homogeneous linear differential equations of order 2 with Galois group or is presented.