On the restricted Hilbert-Speiser and Leopoldt properties

2009/05/17 by Nigel P. Byott, Byott, Nigel P., James E. Carter +5
Mathematics · #11R29 #11R33 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R29 #msc:11R33

paper · pdf · doi:10.48550/arxiv.0905.2737

15 pages, latex, to appear in Illinois Journal of Mathematics.

arxiv created 2010/03/10 · arxiv updated 2010/03/11

Abstract

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O_L is free as an O_K[G]-module. If O_L is free over the associated order A_L/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G=C_p has prime order. We give examples with G=C_p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

Related

• Relative Galois module structure of rings of integers of absolutely Abelian number fields
• The Hilbert-Grunwald specialization property over number fields
• Leopoldt-type theorems for non-abelian extensions of Q
• Specialization results and ramification conditions
• Applications of representation theory and of explicit units to Leopoldt's conjecture