Isomorphism extension theorem
Encyclopedia
In field theory
, a branch of mathematics
, the isomorphism extension theorem is an important theorem
regarding the extension of a field isomorphism
to a larger field.
, an algebraic extension
field
of 
and an isomorphism 
mapping 
onto a field 
then 
can be extended to an isomorphism 
mapping 
onto an algebraic extension 
of 
(a subfield of the algebraic closure
of
).
The proof of the isomorphism extension theorem depends on Zorn's lemma
.
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the isomorphism extension theorem is an important theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
regarding the extension of a field isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to a larger field.
Isomorphism extension theorem
The theorem states that given any field, an algebraic extensionAlgebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
field
of and an isomorphism mapping onto a field then can be extended to an isomorphism mapping onto an algebraic extension of (a subfield of the algebraic closureAlgebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of
).The proof of the isomorphism extension theorem depends on Zorn's lemma
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
.