Editing Splitting field (section)

===The construction===
Let ''F'' be a field and ''p''(''X'') be a polynomial in the [[polynomial ring]] ''F''[''X''] of [[degree of a polynomial|degree]] ''n''. The general process for constructing ''K'', the splitting field of ''p''(''X'') over ''F'', is to construct a [[chain (ordered set)|chain]] of fields <math>F=K_0 \subseteq K_1 \subseteq \cdots \subseteq K_{r-1} \subseteq K_r=K</math> such that ''K<sub>i</sub>'' is an extension of ''K''<sub>''i''−1</sub> containing a new root of ''p''(''X''). Since ''p''(''X'') has at most ''n'' roots the construction will require at most ''n'' extensions. The steps for constructing ''K<sub>i</sub>'' are given as follows:
* [[Factorization of polynomials#Factoring over algebraic extensions (Trager's method)|Factorize]] ''p''(''X'') over ''K<sub>i</sub>'' into [[irreducible polynomial|irreducible]] factors <math>f_1(X)f_2(X) \cdots f_k(X)</math>.
* Choose any nonlinear irreducible factor ''f''(''X'').
* Construct the [[field extension]] ''K''<sub>''i''+1</sub> of ''K<sub>i</sub>'' as the [[quotient ring]] ''K''<sub>''i''+1</sub> = ''K''<sub>''i''</sub>[''X''] / (''f''(''X'')) where (''f''(''X'')) denotes the [[ideal (ring theory)|ideal]] in ''K''<sub>''i''</sub>[''X''] generated by ''f''(''X'').
* Repeat the process for ''K''<sub>''i''+1</sub> until ''p''(''X'') completely factors.

The irreducible factor ''f''(''X'') used in the quotient construction may be chosen arbitrarily. Although different choices of factors may lead to different subfield sequences, the resulting splitting fields will be isomorphic.

Since ''f''(''X'') is irreducible, (''f''(''X'')) is a [[maximal ideal]] of ''K''<sub>''i''</sub>[''X''] and ''K''<sub>''i''</sub>[''X''] / (''f''(''X'')) is, in fact, a field, the [[residue field]] for that maximal ideal. Moreover, if we let <math>\pi : K_i[X] \to K_i[X]/(f(X))</math> be the natural projection of the [[ring (mathematics)|ring]] onto its quotient then
:<math>f(\pi(X)) = \pi(f(X)) = f(X)\ \bmod\ f(X) = 0</math>
so ''π''(''X'') is a root of ''f''(''X'') and of ''p''(''X'').

The degree of a single extension <math>[K_{i+1} : K_i]</math> is equal to the degree of the irreducible factor ''f''(''X''). The degree of the extension [''K'' : ''F''] is given by <math>[K_r : K_{r-1}] \cdots [K_2 : K_1] [K_1 : F]</math> and is at most ''n''!.