A number field is a finite extension of the field 
of rational numbers. Every
number field has the form 
for an algebraic number 
.
It consists of the expressions obtained from 
and rational numbers by
addition, subtraction,
multiplication, and division
by nonzero elements.
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.
Polynomial equations over 
,
such as the quintic equation, can be studied
using the Galois groups of their splitting
fields over 
. Klüners and Malle compiled
a database of number fields classified by the Galois
groups of the splitting fields of their defining
polynomials. For example, the polynomial 
has a splitting
field over 
with Galois group 
of group order 20.
