section_path: Abstract algebraic aspects/Construction as a quotient field
section_name: Construction as a quotient field
paragraph_text: Construction as a quotient field
  
  One approach to C {\displaystyle \mathbb {C} } is via polynomials, i.e., expressions of the form p (X) = a n X n + ⋯ + a 1 X + a 0, {\displaystyle p(X)=a_{n}X^{n}+\dotsb +a_{1}X+a_{0},} where the coefficients a,..., a are real numbers. The set of all such polynomials is denoted by R {\displaystyle \mathbb {R} }. Since sums and products of polynomials are again polynomials, this set R {\displaystyle \mathbb {R} } forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number p (i) = a n i n + ⋯ + a 1 i + a 0 {\displaystyle p(i)=a_{n}i^{n}+\dotsb +a_{1}i+a_{0}}, i.e., the value obtained by setting X = i {\displaystyle X=i}. This defines a function
  
  This function is surjective since every complex number can be obtained in such a way: the evaluation of a linear polynomial a + b X {\displaystyle a+bX} at X = i {\displaystyle X=i} is a + b i {\displaystyle a+bi}. However, the evaluation of polynomial X 2 + 1 {\displaystyle X^{2}+1} at i is 0, since i 2 + 1 = 0. {\displaystyle i^{2}+1=0.} This polynomial is irreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts of abstract algebra then imply that the kernel of the above map is an ideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is an isomorphism
  
  between the quotient ring and C {\displaystyle \mathbb {C} }. Some authors take this as the definition of C {\displaystyle \mathbb {C} }.
  
  Accepting that C {\displaystyle \mathbb {C} } is algebraically closed, because 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}.}