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In abstract algebra, a rupture field of a polynomial P {\displaystyle P} over a given field K {\displaystyle K} is a field extension of K {\displaystyle K} generated by a root a {\displaystyle a} of P {\displaystyle P}.
For instance, if K = Q {\displaystyle K=\mathbb {Q} } and P = X 3 − 2 {\displaystyle P=X^{3}-2} then Q {\displaystyle \mathbb {Q} {2}}]} is a rupture field for P {\displaystyle P}.
The notion is interesting mainly if P {\displaystyle P} is irreducible over K {\displaystyle K}. In that case, all rupture fields of P {\displaystyle P} over K {\displaystyle K} are isomorphic, non-canonically, to K P = K / ] {\displaystyle K_{P}=K/]} : if L = K {\displaystyle L=K} where a {\displaystyle a} is a root of P {\displaystyle P} , then the ring homomorphism f {\displaystyle f} defined by f = k {\displaystyle f=k} for all k ∈ K {\displaystyle k\in K} and f = a {\displaystyle f=a} is an isomorphism. Also, in this case the degree of the extension equals the degree of P {\displaystyle P}.
A rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field Q {\displaystyle \mathbb {Q} {2}}]} does not contain the other two roots of P {\displaystyle P} {2}}} and ω 2 2 3 {\displaystyle \omega ^{2}{\sqrt{2}}} where ω {\displaystyle \omega } is a primitive cube root of unity]. For a field containing all the roots of a polynomial, see Splitting field.