Displaying similar documents to “A generalization of a theorem of Minkowski”

On metacyclic extensions

Masanari Kida (2012)

Journal de Théorie des Nombres de Bordeaux

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Galois extensions with various metacyclic Galois groups are constructed by means of a Kummer theory arising from an isogeny of certain algebraic tori. In particular, our method enables us to construct algebraic tori parameterizing metacyclic extensions.

Finite-dimensional differential algebraic groups and the Picard-Vessiot theory

Anand Pillay (2002)

Banach Center Publications

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We make some observations relating the theory of finite-dimensional differential algebraic groups (the ∂₀-groups of [2]) to the Galois theory of linear differential equations. Given a differential field (K,∂), we exhibit a surjective functor from (absolutely) split (in the sense of Buium) ∂₀-groups G over K to Picard-Vessiot extensions L of K, such that G is K-split iff L = K. In fact we give a generalization to "K-good" ∂₀-groups. We also point out that the "Katz group" (a certain linear...

Groups of Order 32 as Galois Groups

Michailov, Ivo (2007)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 12F12. We find the obstructions to realizability of groups of order 32 as Galois groups over arbitrary field of characteristic not 2. We discuss explicit extensions and automatic realizations as well. This work is partially supported by project of Shumen University

On the inverse problem of Galois theory.

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.

On double covers of the generalized alternating group ℤ d ≀ m as Galois groups over algebraic number fields

Martin Epkenhans (1997)

Acta Arithmetica

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Let ℤ d ≀ m b e t h e g e n e r a l i z e d a l t e r n a t i n g g r o u p . W e p r o v e t h a t a l l d o u b l e c o v e r s o f ℤd ≀ m o c c u r a s G a l o i s g r o u p s o v e r a n y a l g e b r a i c n u m b e r f i e l d . W e f u r t h e r r e a l i z e s o m e o f t h e s e d o u b l e c o v e r s a s t h e G a l o i s g r o u p s o f r e g u l a r e x t e n s i o n s o f ℚ ( T ) . I f d i s o d d a n d m > 7 , t h e n e v e r y c e n t r a l e x t e n s i o n o f ℤd ≀ m o c c u r s a s t h e G a l o i s g r o u p o f a r e g u l a r e x t e n s i o n o f ℚ ( T ) . W e f u r t h e r i m p r o v e s o m e o f o u r e a r l i e r r e s u l t s c o n c e r n i n g d o u b l e c o v e r s o f t h e g e n e r a l i z e d s y m m e t r i c g r o u p ℤd ≀ m .

Galois realizability of groups of order 64

Helen Grundman, Tara Smith (2010)

Open Mathematics

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This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.