?? associated graded object of ideal power filtration of ideal

in z ..... ??? .... ???? ....

p = 5, say ....

?? then consider, say, extension z[i] .... ??? does this come along for the ride in some interesting way ??? .....

?? polynomials over z/5 .... ???? ....

-1 .... ???? .....

?? ideal power filtration of certain (??which??) ideal in gaussian integers .... ????? ......

?? adjoining square root of -1 to f_5[x] ??? .... ??? is that what happens here ??? .... ???? .... ?? how "involved" is x here ??? .... ???? ... ?? maybe not much ?? .... ?? and generally doesn't get any more involved in other cases either ?? .... ???? ... ?? hmmmm ..... ???? ramification ??? .... ?? ideal power filtration of ideal associated with 2 in gaussian integers .... ????? ..... ?? ideal power filtration associated with 5 in golden integers .... ?????? ..... ????? .....

"symbol" ... ???? ... "principal symbol" ??? .... ??? "secondary symbol" ???? ..... ?? "obstruction" ??? ... ??? ... ???? ..... ???? "hessian" ???? .... ???? ..... ??? .....

?? "y^2 = x^3" ..... ???? ......