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-Pierre

| 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=\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.

A 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 cube root of unity). For a field containing all the roots of a polynomial, see Splitting field.

Examples

A rupture field of X^2+1 over \mathbb R is \mathbb C. It is also a 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 which squares to -1 (and all quadratic extensions of \mathbb F_3 are isomorphic to \mathbb F_9).

References

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Category:Field (mathematics)