Rupture field

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In abstract algebra, a rupture field of a polynomial over a given field such that is a field extension of generated by a root of .[1]

For instance, if and then is a rupture field for .

The notion is interesting mainly if is irreducible over . In that case, all rupture fields of over are isomorphic, non canonically, to : if where is a root of , then the ring homomorphism defined by for all and is an isomorphism. Also, in this case the degree of the extension equals the degree of .

A rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field does not contain the other two (complex) roots of (namely and where is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.

Examples[edit]

A rupture field of over is . It is also a splitting field.

The rupture field of over is since there is no element of with square equal to (and all quadratic extensions of are isomorphic to ).

See also[edit]

References[edit]

  1. ^ Escofier, Jean-Paul (2001). Galois Theory. Springer. p. 62. ISBN 0-387-98765-7.