The Fundamental Units Of Pure Cubic Fields

Algebraic number fields are the basic object of research in algebraic number theory.The determination of the fundamental units of number fields have always been one of the most basic classical topics in algebraic number theory.However,it is not easy to calculate the fundamental unit of a number field.In ramification theory,it is found that there is a special relationship between the totally ramified prime number and the fundamental unit in the number field whose class number is not divisible by the degree of the said number field.According to this relation,we give the congruence relations for the fundamental unit of a real quadratic field whose class number is not divisible by 2 and a pure cubic field whose class number is not divisible by 3 with a power integral basis.Furthermore,we study the fundamental unit of a pure cubic field whose class number is not divisible by 3 with no power integral basis.