cyclic extension

A cyclic extension is the simplest possible non-trivial Galois extension: all of its symmetries are powers of a single one. Just as a clock's hours are generated by repeatedly advancing by one, every automorphism here is some iterate of a single generating automorphism. This makes cyclic extensions the atoms from which solvable towers are built.

Formally, a Galois extension L/K is cyclic if Gal(L/K) is a cyclic group, that is, generated by one element. Every cyclic extension is abelian (cyclic groups are abelian), and conversely every finite abelian extension factors, via the structure theorem for finite abelian groups, into a compositum of cyclic pieces. Degree-n cyclic extensions over fields with the n-th roots of unity are exactly the Kummer extensions K(a^{1/n}); without those roots, Artin-Schreier theory handles the characteristic-p case using x^p - x - a.

Cyclic extensions are the rungs of a radical tower: a polynomial is solvable by radicals precisely when its splitting field sits atop a tower of cyclic (in fact, after adjoining roots of unity, Kummer or Artin-Schreier) extensions, equivalently when its Galois group is solvable. Hilbert's Theorem 90 gives the multiplicative description of degree-n cyclic extensions: an element has norm 1 iff it is sigma(b)/b for a generator sigma.