Editing Splitting field (section)

=== Cubic example ===
Let {{mvar|K}} be the [[rational number field]] {{math|'''Q'''}} and {{math|''p''(''x'') {{=}} ''x''<sup>3</sup> − 2}}. Each root of {{mvar|p}} equals {{math|{{radic|2|3}}}} times a [[cube root of unity]]. Therefore, if we denote the cube roots of unity by

:<math>\omega_1 = 1,\,</math> <!-- do not delete "\,": it improves the display of formula on certain browsers. -->
:<math>\omega_2 = -\frac{1}{2} + \frac{\sqrt{3}}{2} i,</math>
:<math>\omega_3 = -\frac{1}{2} - \frac{\sqrt{3}}{2} i.</math>

any field containing two distinct roots of {{mvar|p}} will contain the quotient between two distinct cube roots of unity. Such a quotient is a [[primitive root of unity|primitive]] cube root of unity—either <math>\omega_2</math> or <math>\omega_3=1/\omega_2</math>. It follows that a splitting field {{mvar|L}} of {{mvar|p}} will contain ''ω''<sub>2</sub>, as well as the real [[cube root]] of 2; [[converse (logic)|conversely]], any extension of {{math|'''Q'''}} containing these elements contains all the roots of {{mvar|p}}. Thus

:<math>L = \mathbf{Q}(\sqrt[3]{2}, \omega_2) = \{ a + b\sqrt[3]{2} + c{\sqrt[3]{2}}^2 + d\omega_2 + e\sqrt[3]{2}\omega_2 + f{\sqrt[3]{2}}^2 \omega_2 \mid a,b,c,d,e,f \in \mathbf{Q} \}</math>

Note that applying the construction process outlined in the previous section to this example, one begins with <math>K_0 = \mathbf{Q}</math> and constructs the field <math>K_1 = \mathbf{Q}[X] / (X^3 - 2)</math>. This field is not the splitting field, but contains one (any) root.  However, the polynomial <math>Y^3 - 2</math> is not [[irreducible polynomial|irreducible]] over <math>K_1</math>  and in fact:

:<math>Y^3 -2 = (Y - X)(Y^2 + XY + X^2).</math>

Note that <math>X</math> is not an [[indeterminate (variable)|indeterminate]], and is in fact an element of <math>K_1</math>.  Now, continuing the process, we obtain <math>K_2 = K_1[Y] / (Y^2 + XY + X^2)</math>, which is indeed the splitting field and is spanned by the <math>\mathbf{Q}</math>-basis <math>\{1, X, X^2, Y, XY, X^2 Y\}</math>. Notice that if we compare this with <math>L</math> from above we can identify <math>X = \sqrt[3]{2}</math> and <math>Y = \omega_2</math>.