Essential extension

From formulasearchengine
Revision as of 13:56, 14 November 2013 by 130.235.3.80 (talk) (Properties)
Jump to navigation Jump to search

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

For instance, if K= and P(X)=X32 then [23] 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 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 kK and f(XmodP)=a is an isomorphism. Also, in this case the degree of the extension equals the degree of P.

The rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field [23] does not contain the other two (complex) roots of P(X) (namely ω23 and ω223 where ω is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.

Examples

The rupture field of X2+1 over is . It is also its splitting field.

The rupture field of X2+1 over 𝔽3 is 𝔽9 since there is no element of 𝔽3 with square equal to 1 (and all quadratic extensions of 𝔽3 are isomorphic to 𝔽9).

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  1. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534