algebraic number fields number fields numbers algebra algebraic numbers algebraic number theory
An algebraic number is any complex number that is a solution to some polynomial equation 



f
(
x
)
=
0


{\displaystyle f(x)=0}

 with rational coefficients; for example, every solution 



x


{\displaystyle x}

 of 




x

5


+
(
11

/

2
)

x

3


−
7

x

2


+
9
=
0


{\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0}

 (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.
