 The field

is an extension field of both formula_20 and formula_17 of degree 2 and 4 respectively.It is also a simple extension, as one can show that 

Finite extensions of formula_23 are also called algebraic number fields and are important in number theory.Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers formula_24 for a prime number "p".