In abstract algebra, a rupture field of a polynomial P(X) over a given field K is a field extension of K generated by a root a of P(X). For instance, if K=\mathbb Q and P(X)=X^3-2 then 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 : 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\in K and 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 does not contain the other two (complex) roots of P(X) (namely and where \omega 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 X^2+1 over \mathbb R is \mathbb C. It is also a splitting field. The rupture field of X^2+1 over \mathbb F_3 is \mathbb F_9 since there is no element of \mathbb F_3 which squares to -1 (and all quadratic extensions of \mathbb F_3 are isomorphic to \mathbb F_9).