Number Field

If $r$ is an Algebraic Number of degree $n$, then the totality of all expressions that can be constructed from $r$ by repeated additions, subtractions, multiplications, and divisions is called a number field (or an Algebraic Number Field) generated by $r$, and is denoted $F[r]$. Formally, a number field is a finite extension $\Bbb{Q}(\alpha)$ of the Field $\Bbb{Q}$ of Rational Numbers.

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

$$z^n+a_{n-1}z^{n-1}+\ldots+a_0=0$$

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

See also Algebraic Function Field, Algebraic Integer, Algebraic Number, Field, Finite Field, Q, Quadratic 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.