## Splitting Fields

The quadratic equation does not factorise. This means that it has no fractional roots of the form where a and b are whole numbers. In fact it has no real roots because the discriminant The set of fractions, or rationals, is a field, labelled  has no rational roots.

In fact the roots are and  is rational but is not. We cannot express the roots of the polynomial in terms of the field but if we add to the field we can express the roots in terms of the extended field, written  is called an splitting field of the field Consider the quintic equation This equation factorises into The first factor has root and above and the second has roots and We can't express the roots of this polynomial in terms of rational fractions or even in terms of the rational fractions and but we can express the roots in terms of rational fractions, and This field is written and includes all expressions of the form or where We can always extend to include the roots of a polynomial in this way. We can find a splitting field for any polynomial. 