# Splitting field of a polynomial

The smallest field containing all roots of that polynomial. More exactly, an extension of a field is called the splitting field of a polynomial over the field if decomposes over into linear factors:

and if (see Extension of a field). A splitting field exists for any polynomial , and it is defined uniquely up to an isomorphism that is the identity on . It follows from the definition that a splitting field is a finite algebraic extension of .

Examples. The field of complex numbers serves as the splitting field of the polynomial over the field of real numbers. Any finite field , where , is the splitting field of the polynomial over the prime subfield .