??? confusion about ..... ???question of extent to which arbitrary abelian extensions are simply radical extensions ... ???? well, perhaps simply question of whether roots of unity are already available .... ?? but still... ?? for example, are we claiming that .... ???without assuming all roots of unity already available.... abelian extensions are still radical extensions ... ???? ......

??actually, is that really true??? ..... having roots of unity as needed (??) for radical to imply abelian, but ...... ????? is it maybe also needed for abelian to imply radical ???? .......

???idea that it's disappointing if moduli field of elliptic curve with complex multiplication just amounts to some boring subextension of maximal radical extension .... ???? ..... ???????.....

???hmmm, or was i getting pretty mixed up here .... ??? are radical extensions maybe not all that boring, even after all roots of unity are available ??? .... or maybe they are ?????? ...... ????non-/abelianness of iterated radical extensions ??? .... well, of course we know about "solvable galois group" ..... ????? ......

???"compounded but not iterated radical extensions" ..... ?????? ......

?? "adjoining all radicals of all quantities already in the field" ..... ??????? ...... ??????? ...... ??????? ...........

??? possibility of worse confusion here than already noticing ... ??? .....

???? does "maximal cyclotomic extension" qualify as "boring subextension of maximal non-iterated radical extension" ???? ..... ??? well, perhaps yes ... ??well, certainly seems to be subextension, though maybe could argue about boringness .... roots _of unity_ .... ?????? .....

"p-torsion" ...... ????? roots of ..... ??????? ........

??????maximal abelian extension of cyclotomic extension .... ??? or of maximal cyclotomic extension ???? .... ??????? finding maximal abelian extension of sub-cyclotomic field inside maximal abelian extension of maximal cyclotomic field .... ?????? .......

??? "jacobi reciprocity" ...?? or am i misremembering name here ... ??? something like that as maybe close to encompassing full artin reciprocity ... ?????? .... ????? .......

??? maximal abelian extension of Q inside maximal abelian extension of imaginary quadratuc field .... ??? how it fits inside, in terms of jugendtraum ideas and so forth ... ???? ??? "idomeneal" ??? ..... "genus vs class" ... ???? ......

??? trying to interpret "normal cyclic" as "radical" in _"geometric"_ case .... ???? ....... ???? ..... ??? branch point .... ????? ......