Essential extension

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 $P(X)$.

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 $P$.

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 $\omega {\sqrt[{3}]{2}}$ 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

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 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

• Splitting field

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.