Rupture Field

In abstract algebra, a rupture field of a polynomial over a given field such that is the field extension of generated by a root of .

For instance, if and then is a rupture field for .

The notion is interesting mainly if is irreducible over . In that case, all rupture fields of over are isomorphic, non canonically, to : if where is a root of, then the ring homomorphism defined by for all and is an isomorphism. Also, in this case the degree of the extension equals the degree of .

The rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field does not contain the other two (complex) roots of (namely and where is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.