Integral Basis Of The Algebraic Number Fields

One of the main research objects in algebraic number theory is the algebraic number field,in which people concern the ring of algebraic integers,the decomposition of prime ideals,and so on.In an algebraic number field K,the ring consisting of all algebraic integers is denoted by Ok.Although the integral bases of OK exist,it is quite difficult to specify them.In this paper,we study the construction of the integral bases of cubic fields.The main contents are organized as follows:In chapter 1,we introduce some historical background,research significance and research status about the integral bases of rings of algebraic integers.In chapter 2,we introduce some preliminary for computing the integral bases of rings of algebraic integers.In chapter 3 and 4,we specify the integral bases of the cubic fields by using Okutsu’s method.First,we obtain the irreducible factors of the polynomial f（x）= x³ + a₁x² + a₂x+a₃∈Z[x]in Q_p[x]by means of the Newton diagram and Hensel’s Lemma.Second,we obtain the integral basis of Q_p（θ）for each prime number P,where θ is a root of f（x）.Finally,we construct the integral basis of Q（θ）in the global case.