On a notion of "Galois closure" for extensions of rings

2010/06/13 by Manjul Bhargava, Matthew Satriano, Bhargava, Manjul +1
Mathematics · #11R32 #11R33 #13B05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #math.AC #math.AG #math.NT #msc:11R32 #msc:11R33 #msc:13B05

paper · pdf · doi:10.48550/arxiv.1006.2562

31 pages

arxiv created 2012/08/04 · arxiv updated 2012/08/07

Abstract

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 S_n degree n 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.

Related

• Galois closure data for extensions of rings
• On the characterization of Galois extensions
• The Galois Theory (a ring theoretic approach)
• The Galois closure for rings and some related constructions
• On the transcendental Galois extensions