In abstract algebra, a rupture field of a polynomial P ( X ) over a given field K such that P ( X ) ∈ K [ X ] is a field extension of K generated by a root a of P ( X ) .
For instance, if K = Q and P ( X ) = X 3 − 2 then Q [ 2 3 ] is a rupture field for P ( X ) .
The notion is interesting mainly if P ( X ) is irreducible over K . In that case, all rupture fields of P ( X ) over K are isomorphic, non canonically, to K P = K [ X ] / ( P ( X ) ) : if L = K [ a ] where a is a root of P ( X ) , then the ring homomorphism f defined by f ( k ) = k for all k ∈ K and f ( X mod P ) = a is an isomorphism. Also, in this case the degree of the extension equals the degree of P .
A rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field Q [ 2 3 ] does not contain the other two (complex) roots of P ( X ) (namely ω 2 3 and ω 2 2 3 where ω is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.
A rupture field of X 2 + 1 over R is C . It is also a splitting field.
The rupture field of X 2 + 1 over F 3 is F 9 since there is no element of F 3 with square equal to − 1 (and all quadratic extensions of F 3 are isomorphic to F 9 ).
