三次数域的整基和幂元整基

如何有效地找出数域的整基，是人们长期思考的问题.本文给出了求三次数域的整基的一个简单方法.此外，人们还对数域是否存在幂元整基的问题感兴趣.二次域和分圆域均存在幂元整基，而对于三次数域情形，则尚不清楚.本文给出了三次数域存在幂元整基的充要条件，从而对于三次数域情形给出了完整的答案.

Integral Basis and Power Integral Basis of Cubic Field

How to find out the integral basis of the number field effectively is a problem that people have been thinking for a long time. This paper gives a simple method to find out the integral basis of the cubic field. In addition, people are interested in the existence of power integral basis in number field. There is a power integral basis in both the quadratic field and the cyclotomic field, but it is not clear for the cubic field. In this paper, we give the necessary and sufficient conditions for the existence of power integral basis in the cubic field, and give a complete answer for the case of the cubic field.