Relative Galois module structure of rings of integers of absolutely Abelian number fields

2006/02/27 by Henri Johnston, Johnston, Henri
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Cryptography and Residue Arithmetic #math.NT #msc:11R04 #msc:11R33

paper · pdf · doi:10.48550/arxiv.math/0602637

18 pages, uses xypic. Completely rewritten following referee's report (note that first version contained serious error). To appear in Crelle

arxiv created 2007/07/05 · arxiv updated 2009/12/01

Abstract

Let L/K be an extension of number fields where L/\Q is abelian. We define such an extension to be Leopoldt if the ring of integers O_L of L is free over the associated order A_L/K. Furthermore we define an abelian number field K to be Leopoldt if every finite extension L/K with L/Q abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan & Lim, Bley, and Byott & Lettl culminate in the proof that the n-th cyclotomic field Q^(n) is Leopoldt for every n. In this paper, we generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit generators.

Related

• Leopoldt-type theorems for non-abelian extensions of Q
• On the restricted Hilbert-Speiser and Leopoldt properties
• Applications of representation theory and of explicit units to Leopoldt's conjecture
• Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold
• Nontrivial Galois module structure of cyclotomic fields