In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an algebraic extension 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 ...
such that every element of the larger
field is
algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
over the smaller field ; that is, if every element of is a root of a non-zero
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 exampl ...
with coefficients in . A field extension that is not algebraic, is said to be
transcendental, and must contain
transcendental elements, that is, elements that are not algebraic.
The algebraic extensions of the field
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s are called
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s and are the main objects of study of
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
. Another example of a common algebraic extension is the extension
of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s by the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s.
Some properties
All transcendental extensions are of infinite
degree. This in turn implies that all finite extensions are algebraic. The
converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
s is an infinite algebraic extension of the rational numbers.
Let ''E'' be an extension field of ''K'', and ''a'' ∈ ''E''. If ''a'' is algebraic over ''K'', then ''K''(''a''), the set of all polynomials in ''a'' with coefficients in ''K'', is not only a
ring but a field: ''K''(''a'') is an algebraic extension of ''K'' which has finite degree over ''K''. The converse is not true. Q
�and Q
are fields but π and e are transcendental over Q.
An
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
''F'' has no proper algebraic extensions, that is, no algebraic extensions ''E'' with ''F'' < ''E''. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its
algebraic closure), but
proving this in general requires some form of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.
An extension ''L''/''K'' is algebraic
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
every sub ''K''-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
of ''L'' is a field.
Properties
The following three properties hold:
[Lang (2002) p.228]
# If ''E'' is an algebraic extension of ''F'' and ''F'' is an algebraic extension of ''K'' then ''E'' is an algebraic extension of ''K''.
# If ''E'' and ''F'' are algebraic extensions of ''K'' in a common overfield ''C'', then the
compositum
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...
''EF'' is an algebraic extension of ''K''.
# If ''E'' is an algebraic extension of ''F'' and ''E'' > ''K'' > ''F'' then ''E'' is an algebraic extension of ''K''.
These finitary results can be generalized using transfinite induction:
This fact, together with
Zorn's lemma (applied to an appropriately chosen
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
), establishes the existence of
algebraic closures.
Generalizations
Model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
generalizes the notion of algebraic extension to arbitrary theories: an
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
of ''M'' into ''N'' is called an algebraic extension if for every ''x'' in ''N'' there is a
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
''p'' with parameters in ''M'', such that ''p''(''x'') is true and the set
:
is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of ''N'' over ''M'' can again be defined as the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s, and it turns out that most of the theory of Galois groups can be developed for the general case.
Relative algebraic closures
Given a field ''k'' and a field ''K'' containing ''k'', one defines the relative algebraic closure of ''k'' in ''K'' to be the subfield of ''K'' consisting of all elements of ''K'' that are algebraic over ''k'', that is all elements of ''K'' that are a root of some nonzero polynomial with coefficients in ''k''.
See also
*
Integral element
*
Lüroth's theorem
*
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
*
Separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polyno ...
*
Normal extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic e ...
Notes
References
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{{DEFAULTSORT:Algebraic Extension
Field extensions