A number field is a finite extension of the field rational numbers . Every
number field has the form algebraic number 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 .
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
See also Algebraic Integer ,
Algebraic Number ,
Field ,
Field Norm ,
Finite Field ,
Function
Field ,
Local Field ,
Number
Field Discriminant ,
Number Field Order ,
Number Field Sieve ,
Number
Field Signature ,
Number Ring ,
Primitive
Number Field ,
Q ,
Quadratic
Field ,
Totally Real Field
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References Cohen, H. A Course in Computational Algebraic Number Theory, 3rd. corr. ed. Courant, R. and Robbins, H. What
Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Klüners, J.
and Malle, G. "A Database for Number Fields." http://www.mathematik.uni-kassel.de/~klueners/minimum/minimum.html . The
L-Functions and Modular Forms Database (LMFDB). "Number Fields." https://www.lmfdb.org/NumberField/ . Shanks,
D. Solved
and Unsolved Problems in Number Theory, 4th ed. Wolfram, S. A
New Kind of Science. 1168 ,
2002. Referenced on Wolfram|Alpha Number Field
Cite this as:
Weisstein, Eric W. "Number Field." From
MathWorld https://mathworld.wolfram.com/NumberField.html
Subject classifications
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.
is associated with the group 
of order 20.
