rupture field

In abstract algebra, a rupture field of a polynomial $P(X)$ over a given field $K$ is a field extension of $K$ generated by a root $a$ of $P(X)$.{{Cite book

| last = Escofier

| first = Jean-Paul

| title = Galois Theory

| url = https://archive.org/details/galoistheorygrad00esco

| url-access = limited

| publisher = Springer

| date = 2001

| pages = [https://archive.org/details/galoistheorygrad00esco/page/n75 62]

| isbn = 0-387-98765-7}}

For instance, if $K=\backslash mathbb\; Q$ and $P(X)=X^3-2$ then $\backslash mathbb\; Q[\backslash sqrt[3]2]$ 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 $k\backslash in\; K$ and $f(X\backslash mod\; P)=a$ 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 $\backslash mathbb\; Q[\backslash sqrt[3]2]$ does not contain the other two (complex) roots of $P(X)$ (namely $\backslash omega\backslash sqrt[3]2$ and $\backslash omega^2\backslash sqrt[3]2$ where $\backslash omega$ is a primitive cube root of unity). For a field containing all the roots of a polynomial, see Splitting field.