If 
is an algebraic
number of degree 
,
then the totality of all expressions that can be constructed from 
by repeated additions, subtractions, multiplications, and
divisions is called a number field (or an algebraic number field) generated by 
, and is denoted ![F[r]](/images/equations/NumberField/Inline5.svg)
. Formally, a number field is a finite extension 
of the field 
of rational numbers.
The elements of a number field which are roots of a polynomial
 |
with integer coefficients and leading coefficient 1 are called the algebraic integers of that field.
The coefficients of an algebraic equations such as the quintic equation can be characterized by the groups of their associated number fields.
A database of the groups of number field polynomials is maintained by Klüners
and Malle. For example, the polynomial 
is associated with the group 
of order 20.
