On the p-adic completion of the units of a real abelian number field

TextLet E be the units of a real abelian number field k  . We investigate whether E⊗ZpE⊗Zp is a cyclic Zp[G]Zp[G]-module. If p∤#Gp∤#G, then E⊗ZpE⊗Zp is cyclic. If p|#Gp|#G and G   is cyclic, then the cyclicality of E⊗ZpE⊗Zp depends on the capitulation of ideals through the p  -part of the extension k/Qk/Q among other things. There are cases when p|#Gp|#G and we can unequivocally say that E⊗ZpE⊗Zp is cyclic, for example, if #G=p#G=p. This has application to the annihilation of ideal classes, specifically, we get results analogous to Thaine [6, Theorem 3] but applicable in some non-semi-simple cases.VideoFor a video summary of this paper, please click here or visit http://youtu.be/-JwNNOxzwDI.