Editing Splitting field (section)

=== The field ''K''<sub>''i''</sub>[''X'']/(''f''(''X'')) ===
As mentioned above, the quotient ring ''K''<sub>''i''+1</sub> = ''K''<sub>''i''</sub>[''X'']/(''f''(''X'')) is a field when ''f''(''X'') is irreducible. Its elements are of the form

:<math>c_{n-1}\alpha^{n-1} + c_{n-2}\alpha^{n-2} + \cdots + c_1\alpha + c_0</math>

where the ''c<sub>j</sub>'' are in ''K<sub>i</sub>'' and ''α'' = ''π''(''X''). (If one considers ''K''<sub>''i''+1</sub> as a [[vector space]] over ''K<sub>i</sub>'' then the powers ''α''<sup>&thinsp;''j''</sup> for {{nowrap|0 ≤ ''j'' ≤ ''n''−1}} form a [[Basis (linear algebra)|basis]].)

The elements of ''K''<sub>''i''+1</sub> can be considered as polynomials in ''α'' of degree less than ''n''. Addition in ''K''<sub>''i''+1</sub> is given by the rules for polynomial addition, and multiplication is given by polynomial multiplication modulo ''f''(''X''). That is, for ''g''(''α'') and ''h''(''α'') in ''K''<sub>''i''+1</sub> their product is ''g''(''α'')''h''(''α'') = ''r''(α) where ''r''(''X'') is the remainder of ''g''(''X'')''h''(''X'') when divided by ''f''(''X'') in ''K''<sub>''i''</sub>[''X''].

The remainder ''r''(''X'') can be computed through [[polynomial long division]]; however there is also a straightforward reduction rule that can be used to compute ''r''(''α'') = ''g''(''α'')''h''(''α'') directly. First let

:<math>f(X) = X^n + b_{n-1} X^{n-1} + \cdots + b_1 X + b_0.</math>

The polynomial is over a field so one can take ''f''(''X'') to be [[monic polynomial|monic]] [[without loss of generality]]. Now ''α'' is a root of ''f''(''X''), so

:<math>\alpha^n = -(b_{n-1} \alpha^{n-1} + \cdots + b_1 \alpha + b_0).</math>

If the product ''g''(''α'')''h''(''α'') has a term ''α''<sup>''m''</sup> with {{nowrap|''m'' ≥ ''n''}} it can be reduced as follows:

:<math>\alpha^n\alpha^{m-n} = -(b_{n-1} \alpha^{n-1} + \cdots + b_1 \alpha + b_0) \alpha^{m-n} = -(b_{n-1} \alpha^{m-1} + \cdots + b_1 \alpha^{m-n+1} + b_0 \alpha^{m-n})</math>.

As an example of the reduction rule, take ''K<sub>i</sub>'' = '''Q'''[''X''], the ring of polynomials with [[rational number|rational]] coefficients, and take ''f''(''X'') = ''X''<sup>&thinsp;7</sup> − 2. Let <math>g(\alpha) = \alpha^5 + \alpha^2</math> and ''h''(''α'') = ''α''<sup>3</sup> +1 be two elements of '''Q'''[''X'']/(''X''<sup>&thinsp;7</sup> − 2). The reduction rule given by ''f''(''X'') is ''α''<sup>7</sup> = 2 so

:<math>g(\alpha)h(\alpha) = (\alpha^5 + \alpha^2)(\alpha^3 + 1) = \alpha^8 + 2 \alpha^5 + \alpha^2 = (\alpha^7)\alpha + 2\alpha^5 + \alpha^2 = 2 \alpha^5 + \alpha^2 + 2\alpha.</math>