In abstract algebra, a rupture field of a polynomial
P
(
X
)
over a given field
K
such that
P
(
X
)
∈
K
[
X
]
is a field extension of
K
generated by a root
a
of
P
(
X
)
.
For instance, if
K
=
Q
and
P
(
X
)
=
X
3
−
2
then
Q
[
2
3
]
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
∈
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
Q
[
2
3
]
does not contain the other two (complex) roots of
P
(
X
)
(namely
ω
2
3
and
ω
2
2
3
where
ω
is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.
A rupture field of
X
2
+
1
over
R
is
C
. It is also a splitting field.
The rupture field of
X
2
+
1
over
F
3
is
F
9
since there is no element of
F
3
with square equal to
−
1
(and all quadratic extensions of
F
3
are isomorphic to
F
9
).
