In abstract algebra, a rupture field of a polynomial over a given field such that is the field extension of generated by a root of .
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 .
The 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.
Famous quotes containing the words rupture and/or field:
“Awareness requires a rupture with the world we take for granted; then old categories of experience are called into question and revised.”
—Shoshana Zuboff (b. 1951)
“A field of water betrays the spirit that is in the air. It is continually receiving new life and motion from above. It is intermediate in its nature between land and sky.”
—Henry David Thoreau (1817–1862)