Global field

In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:^[1]

• Algebraic number field: A finite extension of ${\displaystyle \mathbb {Q} }$
• Global function field: The function field of an irreducible algebraic curve over a finite field, equivalently, a finite extension of ${\displaystyle \mathbb {F} _{q}(T)}$, the field of rational functions in one variable over the finite field with ${\displaystyle q=p^{n}}$ elements.

An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.^[2][3]