algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety circular units; Jacobian of Fermat curves; Galois representations; pro-\(\ell\) braid groups; étale covering of projective 1-space; representation in outer automorphism group of profinite fundamental; group; absolute Galois group; completed group algebra; Tate module; Jacobi sums; Galois cohomology 