Rupture field

From Wikipedia, the free encyclopedia

In abstract algebra, a rupture field of a polynomial P(X) over a given field K such that P(X)\in K[X] is the field extension of K generated by a root a of 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 KP = 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\in K and f(X\mod P)=a is an isomorphism.

For instance, if K=\mathbb Q and P(X) = X3 − 2 then \mathbb Q[\sqrt[3]2] is a rupture field for P(X).

The rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field \mathbb Q[\sqrt[3]2] does not contain the other two (complex) roots of P(X) (namely j\sqrt[3]2 and j^2\sqrt[3]2 where j is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.

[edit] Examples

The rupture field of X2 + 1 over \mathbb R is \mathbb C. It is also its splitting field.

The rupture field of X2 + 1 over \mathbb F_3 is \mathbb F_9 since there is not element of \mathbb F_3 with square equal to − 1 (and all quadratic extensions of \mathbb F_3 are isomorphic to \mathbb F_9).

[edit] See also

Languages