Characterizing Cyclic Cubic Extensions by Automorphism Polynomials Original Research Article

The arithmetic of iterated maps is used to characterize the cyclic cubic extensions F of a field κ (char κ ≠ 2) in terms of the polynomials representing the nontrivial automorphisms of F/κ. This leads to an analogue of Kummer theory for abelian extensions of exponent 3 of κ, whether or not κ contains a primitive cube root of unity. Such extensions are shown to be in 1-1 correspondence with certain groups of linear fractional transformations defined over κ.