The extension field of a field is called a splitting field for the polynomial if factors completely into linear factors in and does not factor completely into linear factors over any
proper subfield of containing (Dummit and Foote 1998, p. 448).

For example, the extension field is the splitting field for since it is the smallest field containing its roots, and . Note that it is also the splitting field for .

Portions of this entry contributed by Todd
Rowland