Integral Basis And Power Basis Of An Algebraic Number Field

Letω1,...,ω2∈Ok,ifOk=Zω1(?)...(?)Zωn,thenω1,...,ωn is called an integral basis for integral domains OK or algebraic number field k. In other words,ω1,...,ωn is an integral basis of OK or K if and only if each integerα∈Ok can be expressed asα=λ1ω1+...+λnωn∈Z.This paper introduces the integral basis and power basis of an algebraic number field, especially in some cubic fields and composite fields.How to find an integral basis of an algebraic number field has long been concerned. The integral basis of quadratic field and cyclotomic field have been solved completely, the integral basis of pure cubic field has also been given, but the integral basis of the general cubic field is not clear, even for cycle cubic field. The integral basis of composite field is more complex. The second chapter in my paper gives some simple methods for finding the integral basis of some cubic fields and composite fields.In addition, some special types of an integral basis are interested by people, such as power basis. We already know that quadratic field and cyclotomic field possess a power basis, but not all of the algebraic number field possess a power basis. The problem that which algebraic number field has a power basis is also unclear even for cycle cubic fields. The third chapter in my paper gives some simple methods for finding a power basis through some particular examples, and introduces an example of an algebraic number field without a power basis which is given by Dedekind in 1878.