The nonexistence of certain free pro-p extensions and capitulation in a family of dihedral extensions of Q

Abstract

$\doubz\sbsp{p}{d}$-extensions are a natural way to extend Iwasawa's theory of $\doubz\sb{p}$-extensions. A further extension would be to look at free pro-p extensions which though nonabelian, can be studied by looking at their maximal abelian subextension, which is a $\doubz\sbsp{p}{d}$-extension. There is a natural upper bound given by Leopoldt's conjecture for the size of a free pro-p extension of a given number field. We show in the first problem a number field which does not have a free pro-p extension whose size attains this natural upper bound. Although some examples of such number fields are already known, to my knowledge techniques of Iwasawa theory have not been used before to show such a result and should be able to show further insight and examples.We also make a study of capitulation by computer in a family of $S\sb3$- extensions of $\doubq$. We compute class groups as well as capitulation and find a variety of behavior. We give proofs for a number of the observations seen. This appears to be the first set of computations done of this type.