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.

See also

Algebraic Integer, Algebraic Number, Class Field, 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

Explore with Wolfram|Alpha

References

Cohen, H. A Course in Computational Algebraic Number Theory, 3rd. corr. ed. New York: Springer-Verlag, 1996.Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, p. 127, 1996.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. New York: Chelsea, pp. 151-152, 1993.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

Referenced on Wolfram|Alpha

Number Field

Cite this as:

Weisstein, Eric W. "Number Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NumberField.html

Subject classifications