Rupture field explained

P(X) over a given field K is a field extension of K generated by a root a of P(X) .^[1]

For instance, if

K = Q and P(X) = X ^{3-2} then Q[\sqrt[3]2] 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\in 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[\sqrt[3]2] does not contain the other two (complex) roots of P(X) (namely \omega\sqrt[ 3 ] 2 and \omega ^{2\sqrt[3]2} 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 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} which squares to -1 (and all quadratic extensions of F _{3} are isomorphic to F _{9} ).

Notes and References

1. Book: Escofier , Jean-Paul . Galois Theory . limited . Springer . 2001 . 62 . 0-387-98765-7.