On $p-$Rationality of Cubic and Quartic Number Fields

In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are constructed,the relations between generalized $abc-$conjecture and the $p-$rationality are discussed, from which some explicit fields satisfying Greenberg’s Generalized Conjecture (GGC, for short) are obtained.