Rupture field

In abstract algebra, a rupture field of a polynomial ${\displaystyle P(X)}$ over a given field ${\displaystyle K}$ is a field extension of ${\displaystyle K}$ generated by a root ${\displaystyle a}$ of ${\displaystyle P(X)}$.^[1]

For instance, if ${\displaystyle K=\mathbb {Q} }$ and ${\displaystyle P(X)=X^{3}-2}$ then ${\displaystyle \mathbb {Q} [{\sqrt[{3}]{2}}]}$ is a rupture field for ${\displaystyle P(X)}$.

The notion is interesting mainly if ${\displaystyle P(X)}$ is irreducible over ${\displaystyle K}$. In that case, all rupture fields of ${\displaystyle P(X)}$ over ${\displaystyle K}$ are isomorphic, non-canonically, to ${\displaystyle K_{P}=K[X]/(P(X))}$: if ${\displaystyle L=K[a]}$ where ${\displaystyle a}$ is a root of ${\displaystyle P(X)}$, then the ring homomorphism ${\displaystyle f}$ defined by ${\displaystyle f(k)=k}$ for all ${\displaystyle k\in K}$ and ${\displaystyle f(X\mod P)=a}$ is an isomorphism. Also, in this case the degree of the extension equals the degree of ${\displaystyle P}$.

A rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field ${\displaystyle \mathbb {Q} [{\sqrt[{3}]{2}}]}$ does not contain the other two (complex) roots of ${\displaystyle P(X)}$ (namely ${\displaystyle \omega {\sqrt[{3}]{2}}}$ and ${\displaystyle \omega ^{2}{\sqrt[{3}]{2}}}$ where ${\displaystyle \omega }$ 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 ${\displaystyle X^{2}+1}$ over ${\displaystyle \mathbb {R} }$ is ${\displaystyle \mathbb {C} }$. It is also a splitting field.

The rupture field of ${\displaystyle X^{2}+1}$ over ${\displaystyle \mathbb {F} _{3}}$ is ${\displaystyle \mathbb {F} _{9}}$ since there is no element of ${\displaystyle \mathbb {F} _{3}}$ which squares to ${\displaystyle -1}$ (and all quadratic extensions of ${\displaystyle \mathbb {F} _{3}}$ are isomorphic to ${\displaystyle \mathbb {F} _{9}}$).

References