- Rupture field
In
abstract algebra , a rupture field of apolynomial P(X) over a given field K such that P(X)in K [X] is thefield 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 K_P=K [X] /(P(X)): if L=K [a] where a is a root of P(X), then thering homomorphism f defined by f(k)=k for all kin K and f(Xmod P)=a is anisomorphism .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 rupture field of a
polynomial does not necessarily contain all the roots of thatpolynomial : in the above example the field mathbb Q [sqrt [3] 2] does not contain the other two (complex) roots of P(X) (namely jsqrt [3] 2 and j^2sqrt [3] 2 where j is a primitive third root of unity). For a field containing all the roots of apolynomial , see thesplitting field .Examples
The rupture field of X^2+1 over mathbb R is mathbb C. It is also its
splitting field .The rupture field of X^2+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).
ee also
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Splitting field
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