Global field

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a global field is one of two types of <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> (the other one is <a href="/facts/Local_field/vqIgPIX3">local fields</a>) that are characterized using <a href="/facts/Valuation_(algebra)/FXadrz5c">valuations</a>. There are two kinds of global <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>:
</p>
<ul><li><a href="/facts/Algebraic_number_field/e0K0TXSh">Algebraic number field</a>: A <a href="/facts/Degree_of_a_field_extension/hY5LgHYk">finite extension</a> of 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
</li>
<li>Global function field: The <a href="/facts/Function_field_of_an_algebraic_variety/54cc62wP">function field</a> of an <a href="/facts/Irreducible_space/Yu2eFIon">irreducible</a> <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a>, equivalently, a finite extension of 
  
    
      
        
          
            F
          
          
            q
          
        
        (
        T
        )
      
    
    {\displaystyle \mathbb {F} _{q}(T)}
  
, the field of <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> in one variable over the finite field with 
  
    
      
        q
        =
        
          p
          
            n
          
        
      
    
    {\displaystyle q=p^{n}}
  
 elements.</li></ul>
<p>An axiomatic characterization of these fields via <a href="/facts/Valuation_(algebra)/FXadrz5c">valuation theory</a> was given by <a href="/facts/Emil_Artin/rI3vIFo2">Emil Artin</a> and George Whaples in the 1940s.
</p>

Global field open-in-new

Global field