Rational variety

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a rational variety is an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a>, over a given <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K, which is <a href="/facts/Birationally_equivalent/hsWqRhJ1">birationally equivalent</a> to a <a href="/facts/Projective_space/7MSpA9cM">projective space</a> of some dimension over K. This means that its <a href="/facts/Function_field_of_an_algebraic_variety/54cc62wP">function field</a> is isomorphic to

K
        (
        
          U
          
            1
          
        
        ,
        …
        ,
        
          U
          
            d
          
        
        )
        ,
      
    
    {\displaystyle K(U_{1},\dots ,U_{d}),}

the field of all <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a> for some set 
 
 
 
 {
 
 U
 
 1
 
 
 ,
 …
 ,
 
 U
 
 d
 
 
 }
 
 
 {\displaystyle \{U_{1},\dots ,U_{d}\}}
 
 of <a href="/facts/Indeterminate_(variable)/R7QAHS4j">indeterminates</a>, where d is the <a href="/facts/Dimension_of_an_algebraic_variety/sXqIHcYc">dimension</a> of the variety.

Rational variety open-in-new

Rational variety