Abstract
Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These invariants have proved to be a valuable information for determining the Galois group of the second Hilbert p-class field and the p-capitulation type of K. For p=3 and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group of the maximal unramified pro-p extension of K.
Highlights
Before we turn to applications in extreme computing, that is, squeezing the computational algebra systems PARI/GP [1] and MAGMA [2]-[4] to their limits in Section 5, where we show how to detect malformed index- p abelianization data (IPAD) in Section 5.1, and how to complete partial p -capitulation types in Section 5.2, we have to establish a componentwise correspondence between transfer kernel types (TKTs) and IPADs in Section 4 by exploiting details
Since the abelian type invariants of the members of target types (TTTs) layers will depend on the parity of the nilpotency class c or coclass r, a more economic notation, avoiding the tedious distinction of the cases odd or even, is provided by the following definition ([5], Section 3)
441), when we perform a transformation from the first layer TKT and TTT
Summary
These main theorems give all possible IPADs of number fields K with 3-class group Cl3 ( K ) of type (3,3). We emphasize that IPADs of infinite p -class towers reveal an unknown wealth of possible fine structure in Section 7 on complex quadratic fields K having a 3-class group Cl3 ( K ) of type (3,3,3)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
