?? x^3 - x = x^2 .... x^2 - 1 = x ?? ...

?? "inflection point" ?? ...

??so ... ??try to understand inflection points in general here ??? ....

??second google hit on "eight inflection points on" is about "elliptic curves with isomorphic 3-torsion over Q" .... ?????? ......

??so ... ?? artin reciprocity tells us galois group of 3-maximal abelian extension of imaginary quadratic number field of discriminant d (?? ...) is ... ???? ....

??multiplicative group of imaginary quadratic number field f, mod n .... ????? ....

?? arbitrary ring r as algebraic ring given by functor taking commutative ring x to ring r tensor x ???? .... (limit-preservation properties of such functor, in general ??? .... ??? or in less general ?? ...???? ... ???? distributivity of cartesian product of affine schemes over .... ??? finite colimits of affine schemes ... = finite limits of commutative rings ... ???? .... ???? .... ???? .... ????? .....)

?? then giving rise to algebraic group by taking multiplicative group ... ???

?? then specializing to case x = Z/n ???? .....

?? commutativity of tensor product of commutative rings as maybe giving some sort of "reciprocity" here ??? ..... ???? .....

??what _is_ going on here ??? ... ??? any lawvere-theory morphism t -> t_[comm ring] (??how crucial is comm here ???) as giving nice algebraic ... ??? maybe level (??) slip ??? .... mult gp of any (comm ?) ring as nice alg group ... ??? ... more general ... ???? ..... ??? ... ... ??? ....

??actually maybe "flatness" issues here ??? ..... ????? .....

?? hmm, yes ... ??? ... and ring r need not be commutative .... ???? .....

??? multiplicative group of flat ring as nice ( = affine ??) algebraic group ... ???

?? in non-flat case ... ?? not so nice algebraic group ??? .... ??? maybe "stacky" and / or .... ???? ..... ??? maybe somewhat different versions depending on .... ????? ..... ????? ...... ??? "coarse vs fine" ??? .... ???? ..... ?? maybe "higher-affine" sometimes ... ??? ....

??well, so what about "multiplicative group of Z/n" as attempted algebraic group here ??? .... ??? with some sort of "correction" ... ??? ....