Editing Algebraic function field (section)

==Number fields and finite fields==
The [[function field analogy]] states that almost all theorems on [[number field]]s have a counterpart on function fields of one variable over a [[finite field]], and these counterparts are frequently easier to prove. (For example, see [[Prime number theorem#Analogue for irreducible polynomials over a finite field|Analogue for irreducible polynomials over a finite field]].) In the context of this analogy, both number fields and function fields over finite fields are usually called "[[global field]]s".

The study of function fields over a finite field has applications in [[cryptography]] and [[error correcting code]]s. For example, the function field of an [[elliptic curve]] over a finite field (an important mathematical tool for [[public key cryptography]]) is an algebraic function field.

Function fields over the field of [[rational number]]s play also an important role in solving [[inverse Galois problem]]s.