I think you can remove reference to "Hensel" and replace by "Pascal". I.e. all you really need is to work with a very simple application of the binomial theorem, to solve for c by squaring (r + c.prime power) modulo the next prime power up, and equating to 1.

Assuming the "worst" case of N the product of two odd primes, the differences in pairs of the four roots are six numbers, and you can factorise N easily by taking the HCF with three of them ... don't think there is anything here.

@Todd Trimble: the roots of the equation can all be deduced.

I'm not quite sure where you think the "conjectures" were. In any case I like the thought that the real point was that this was the emergence of the "classical group" concept, which was probably not equated with a subset of Lie group theory.

I kind of tried this with the question mathoverflow.net/questions/59827/… . The observation underlying is surely that there is a "Galois connection" between the sets of theorems in geometries, and the (sub)groups determining the geometry. Mathematics of the 19th century was discursive, not axiomatic, so you don't always get the precision.

OK, the representation theory of the infinite dihedral group over a field is a special case, in that the group ring is really twisted Laurent polynomials. You have to assume X acts invertibly, in other words.

Your starting point seems a bit naive, even for the infinite dihedral group and its group ring.

It's a real question, but the answer is clearly "yes". The interesting question is actually the complementary one: what else "survived" the post-1945 rethinking of graduate education in mathematics?

I am no longer a mathematician in fact. I think I can see this "issue" from both sides.