basis of ideal in algebraic number field
where the ’s run all rational integers, form precisely all numbers of . One has also
i.e. the basis of the ideal can be taken for the system of generators of the ideal.
It may be proven that all bases of an ideal have the same discriminant , which is an integer; it is called the discriminant of the ideal. The discriminant of the ideal has the minimality property, that if are some elements of , then
But if , then also the ’s form a basis of the ideal .
Example. The integers of the quadratic field are with . Determine a basis and the discriminant of the ideal a) , b) .
a) The ideal may be seen to be the principal ideal , since the both generators are of the form and on the other side, . Accordingly, any element of the ideal are of the form
where and are rational integers. Thus we can infer that
is a basis of the ideal concerned. So its discriminant is
b) All elements of the ideal have the form
Especially the rational integers of the ideal satisfy , when and thus . This means that in the presentation we can assume to be . Now the rational portion in the form (1) of should be splitted into two parts so that the first would be always divisible by 7 and the second by , i.e. ; this equation may be written also as
By experimenting, one finds the simplest value , another would be . The first of these yields
i.e. we have the basis
The second alternative similarly would give
For both alternatives, .
|Title||basis of ideal in algebraic number field|
|Date of creation||2013-03-22 17:51:15|
|Last modified on||2013-03-22 17:51:15|
|Last modified by||pahio (2872)|
|Synonym||basis of ideal in number field|
|Defines||basis of ideal|
|Defines||discriminant of the ideal|