Rupture field

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In abstract algebra, a rupture field of a polynomial P(X) over a given field K such that P(X)\in K[X] is a field extension of K generated by a root a of P(X).[1]

For instance, if K=\mathbb Q and P(X)=X^3-2 then \mathbb 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 \mathbb 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 third root of unity). For a field containing all the roots of a polynomial, see the splitting field.

Examples[edit]

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 with square equal to -1 (and all quadratic extensions of \mathbb F_3 are isomorphic to \mathbb F_9).

See also[edit]

References[edit]

  1. ^ Escofier, Jean-Paul (2001). Galois Theory. Springer. p. 62. ISBN 0-387-98765-7.