Editing Complex number (section)

===Construction as a quotient field===
One approach to <math>\C</math> is via [[polynomial]]s, i.e., expressions of the form
<math display=block>p(X) = a_nX^n+\dotsb+a_1X+a_0,</math>
where the [[coefficient]]s {{math|''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}} are real numbers. The set of all such polynomials is denoted by <math>\R[X]</math>. Since sums and products of polynomials are again polynomials, this set <math>\R[X]</math> forms a [[commutative ring]], called the [[polynomial ring]] (over the reals). To every such polynomial ''p'', one may assign the complex number <math>p(i) = a_n i^n + \dotsb + a_1 i + a_0</math>, i.e., the value obtained by setting <math>X = i</math>. This defines a function
:<math>\R[X] \to \C</math> 
This function is [[surjective]] since every complex number can be obtained in such a way: the evaluation of a [[linear polynomial]] <math>a+bX</math> at <math>X = i</math> is <math>a+bi</math>. However, the evaluation of polynomial <math>X^2 + 1</math> at ''i'' is 0, since <math>i^2 + 1 = 0.</math> This polynomial is [[irreducible polynomial|irreducible]], i.e., cannot be written as a product of two linear polynomials. Basic facts of [[abstract algebra]] then imply that the [[Kernel (algebra)|kernel]] of the above map is an [[ideal (ring theory)|ideal]] generated by this polynomial, and that the quotient by this ideal is a field, and that there is an [[isomorphism]] 
:<math>\R[X] / (X^2 + 1) \stackrel \cong \to \C</math>
between the quotient ring and <math>\C</math>. Some authors take this as the definition of <math>\C</math>.<ref>{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref>

Accepting that <math>\Complex</math> is algebraically closed, because it is an [[algebraic extension]] of <math>\mathbb{R}</math> in this approach, <math>\Complex</math> is therefore the [[algebraic closure]] of <math>\R.</math>