algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Galois theory; Galois group; full automorphism group; Galois extension; inverse Galois problem; separably generated extension; solvable group; algebraic function field; rational function field; ramified; perfect field; genus; place; Castelnuovo-Severi inequality 