Rupture field
In abstract algebra, a rupture field of a polynomial  over a given field  is a 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 . A 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 cube root of unity). For a field containing all the roots of a polynomial, see Splitting field. Examples
A rupture field of  over  is . It is also a splitting field. The rupture field of  over  is  since there is no element of  which squares to  (and all quadratic extensions of  are isomorphic to ). See also
 Splitting field

References

Field (mathematics)