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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, a rupture field of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
P(X) over a given field K is a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ' ...
of K generated by a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
a of P(X). For instance, if K=\mathbb Q and P(X)=X^3-2 then \mathbb Q sqrt[3.html"_;"title=".html"_;"title="sqrt[3">sqrt[3">.html"_;"title="sqrt[3">sqrt[3/math>_is_a_rupture_field_for_P(X). The_notion_is_interesting_mainly_if_P(X)_is_irreducible_polynomial.html" ;"title="">sqrt[3.html" ;"title=".html" ;"title="sqrt[3">sqrt[3">.html" ;"title="sqrt[3">sqrt[3/math> is a rupture field for P(X). The notion is interesting mainly if P(X) is irreducible polynomial">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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. 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.html"_;"title=".html"_;"title="sqrt[3">sqrt[3">.html"_;"title="sqrt[3">sqrt[3/math>_does_not_contain_the_other_two_(complex_number.html" "title="">sqrt[3.html" ;"title=".html" ;"title="sqrt[3">sqrt[3">.html" ;"title="sqrt[3">sqrt[3/math> does not contain the other two (complex number">complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
) roots of P(X) (namely \omega\sqrt[3]2 and \omega^2\sqrt[3]2 where \omega is a primitive root of unity, primitive cube root of unity). For a field containing all the roots of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
, see
Splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
.


Examples

A rupture field of X^2+1 over \mathbb R is \mathbb C. It is also a
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
. 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 In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ...
to -1 (and all
quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ' ...
s of \mathbb F_3 are isomorphic to \mathbb F_9).


See also

*
Splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...


References

{{DEFAULTSORT:Rupture Field Field (mathematics)