algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety finite fields; number theory; algebraic number theory; computer science; coding theory; cryptography; algebraic geometry; discrete mathematics; polynomial factorization; counting points on curves; irreducible polynomials; primitive polynomials; bases; discrete logarithm; polynomial multiplication; algebraic curves; exponential sums; elliptic curves; recurrence sequences; cyclic codes; pseudo-random numbers; permutation polynomials; congruences; integer factorization; primality testing; computational algebraic number theory; algebraic complexity theory; polynomials with integer coefficients 