This is a term I created for where a cyclic extension of number fields of odd prime p degree takes the place of a quadratic extension and there is also a Galois element tau of order dividing p-1 acting on the extension field.

Many interesting things happen in this context.

For instance there is a sequence of higher ambiguous groups and tau acts on them in a well-defined way with a certain change of action on each successive higher ambiguous group.

There is also a similar action on units mod p-powers.

Eizi Inaba has given an interesting computational method of computing the higher ambiguous groups.

Inaba, E., Uber die Struktur der l-Klassengruppe zyklischer Zahlkorper von Primzahlgrad l, 1944