Complex number no relation a0
where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set 
  
    
      
        
          R
        
        [
        X
        ]
      
    
    {\displaystyle \mathbb {R} [X]}
   of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.
The set of complex numbers is defined as the quotient ring 
  
    
      
        
          R
        
        [
        X
        ]
        
          /
        
        (
        
          X
          
            2
          
        
        +
        1
        )
        .
      
    
    {\displaystyle \mathbb {R} [X]/(X^{2}+1).}
   This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. (The cosets of) 1 and X form a basis of 
  
    
      
        
          R
        
        [
        X
        ]
        
          /
        
        (
        
          X
          
            2
          
        
        +
        1
        )
      
    
    {\displaystyle \mathbb {R} [X]/(X^{2}+1)}
   as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers.  The quotient ring is a field, because X2 + 1 is irreducible over 
  
    
      
        
          R
        
        ,
      
    
    {\displaystyle \mathbb {R} ,}
   so the ideal it generates is maximal.
The formulas for addition and multiplication in the ring 
  
    
      
        
          R
        
        [
        X
        ]
        ,
      
    
    {\displaystyle \mathbb {R} [X],}
   modulo the relation X2 = −1, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs.  So the two definitions of the field 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
   are isomorphic (as fields).
Accepting that 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
   is algebraically closed, since it is an algebraic extension of 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
   in this approach, 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
   is therefore the algebraic closure of 
  
    
      
        
          R
        
        .
      
    
    {\displaystyle \mathbb {R} .}
no relation