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 . Formally, a number field is a finite extension of the Field of Rational Numbers.

The numbers of a number field which are Roots of a Polynomial

with integral coefficients and leading coefficient 1 are called the Algebraic Integers of that field.

**References**

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.

Shanks, D. *Solved and Unsolved Problems in Number Theory, 4th ed.* New York: Chelsea, pp. 151-152, 1993.

© 1996-9

1999-05-25