Rupture field

In abstract algebra, a rupture field of a polynomial formula_1 over a given field formula_2 such that formula_3 is a field extension of formula_2 generated by a root formula_5 of formula_1.
For instance, if formula_7 and formula_8 then formula_9 is a rupture field for formula_1.
The notion is interesting mainly if formula_1 is irreducible over formula_2. In that case, all rupture fields of formula_1 over formula_2 are isomorphic, non canonically, to formula_15: if formula_16 where formula_5 is a root of formula_1, then the ring homomorphism formula_19 defined by formula_20 for all formula_21 and formula_22 is an isomorphism. Also, in this case the degree of the extension equals the degree of formula_23.
A rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field formula_9 does not contain the other two (complex) roots of formula_1 (namely formula_26 and formula_27 where formula_28 is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.
Examples.
A rupture field of formula_29 over formula_30 is formula_31. It is also a splitting field.
The rupture field of formula_29 over formula_33 is formula_34 since there is no element of formula_33 with square equal to formula_36 (and all quadratic extensions of formula_33 are isomorphic to formula_34).