Rupture field
In abstract algebra, a rupture field of a polynomial over a given field such that 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 roots of . For a field containing all the roots of a polynomial, see the 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 with square equal to .
