HOME

TheInfoList



OR:

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 ...
, particularly in
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 ...
, 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 ''F''. For example, under the usual notions of addition and multiplication, 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 are an extension field 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; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.


Subfield

A subfield K of a field L is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under the operations of addition, subtraction, multiplication, and taking the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
of a nonzero element of K. As , the latter definition implies K and L have the same zero element. For example, the field of
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 is a subfield 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, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is
isomorphic 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 ...
to) a subfield of any field of characteristic 0. The characteristic of a subfield is the same as the characteristic of the larger field.


Extension field

If ''K'' is a subfield of ''L'', then ''L'' is an extension field or simply extension of ''K'', and this pair of fields is a field extension. Such a field extension is denoted ''L'' / ''K'' (read as "''L'' over ''K''"). If ''L'' is an extension of ''F'', which is in turn an extension of ''K'', then ''F'' is said to be an intermediate field (or intermediate extension or subextension) of ''L'' / ''K''. Given a field extension , the larger field ''L'' is a ''K''-
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of this vector space is called the degree of the extension and is denoted by 'L'' : ''K'' The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree. Given two extensions and , the extension is finite if and only if both and are finite. In this case, one has : : K : Lcdot : K Given a field extension ''L'' / ''K'' and a subset ''S'' of ''L'', there is a smallest subfield of ''L'' that contains ''K'' and ''S''. It is the intersection of all subfields of ''L'' that contain ''K'' and ''S'', and is denoted by ''K''(''S''). One says that ''K''(''S'') is the field ''generated'' by ''S'' over ''K'', and that ''S'' is a generating set of ''K''(''S'') over ''K''. When S=\ is finite, one writes K(x_1, \ldots, x_n) instead of K(\), and one says that ''K''(''S'') is over ''K''. If ''S'' consists of a single element ''s'', the extension is called a simple extension and ''s'' is called a primitive element of the extension. An extension field of the form is often said to result from the ' of ''S'' to ''K''. In characteristic 0, every finite extension is a simple extension. This is the primitive element theorem, which does not hold true for fields of non-zero characteristic. If a simple extension is not finite, the field ''K''(''s'') is isomorphic to the field of rational fractions in ''s'' over ''K''.


Caveats

The notation ''L'' / ''K'' is purely formal and does not imply the formation of a quotient ring or quotient group or any other kind of division. Instead the slash expresses the word "over". In some literature the notation ''L'':''K'' is used. It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. ''Every'' non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields. Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.


Examples

The field of complex numbers \Complex is an extension field of the field of
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 \R, and \R in turn is an extension field of the field of rational numbers \Q. Clearly then, \Complex/\Q is also a field extension. We have Complex:\R=2 because \ is a basis, so the extension \Complex/\R is finite. This is a simple extension because \Complex = \R(i). R:\Q=\mathfrak c (the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...
), so this extension is infinite. The field :\Q(\sqrt) = \left \, is an extension field of \Q, also clearly a simple extension. The degree is 2 because \left\ can serve as a basis. The field :\begin \Q\left(\sqrt, \sqrt\right) &= \Q \left(\sqrt\right) \left(\sqrt\right) \\ &= \left\ \\ &= \left\, \end is an extension field of both \Q(\sqrt) and \Q, of degree 2 and 4 respectively. It is also a simple extension, as one can show that :\begin \Q(\sqrt, \sqrt) &= \Q (\sqrt + \sqrt) \\ &= \left \. \end Finite extensions of \Q are also called algebraic number fields and are important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers \Q_p for a prime number ''p''. It is common to construct an extension field of a given field ''K'' as a quotient ring of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''K'' 'X''in order to "create" 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 su ...
for a given polynomial ''f''(''X''). Suppose for instance that ''K'' does not contain any element ''x'' with ''x''2 = −1. Then the polynomial X^2+1 is irreducible in ''K'' 'X'' consequently the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
generated by this polynomial is maximal, and L = K (X^2+1) is an extension field of ''K'' which ''does'' contain an element whose square is −1 (namely the residue class of ''X''). By iterating the above construction, one can construct a splitting field of any polynomial from ''K'' 'X'' This is an extension field ''L'' of ''K'' in which the given polynomial splits into a product of linear factors. If ''p'' is any
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
and ''n'' is a positive integer, we have a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
GF(''pn'') with ''pn'' elements; this is an extension field of the finite field \operatorname(p) = \Z/p\Z with ''p'' elements. Given a field ''K'', we can consider the field ''K''(''X'') of all
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in the variable ''X'' with coefficients in ''K''; the elements of ''K''(''X'') are fractions of two
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 ...
s over ''K'', and indeed ''K''(''X'') is the field of fractions of the polynomial ring ''K'' 'X'' This field of rational functions is an extension field of ''K''. This extension is infinite. Given a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
''M'', the set of all meromorphic functions defined on ''M'' is a field, denoted by \Complex(M). It is a transcendental extension field of \Complex if we identify every complex number with the corresponding constant function defined on ''M''. More generally, given an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''V'' over some field ''K'', then the function field of ''V'', consisting of the rational functions defined on ''V'' and denoted by ''K''(''V''), is an extension field of ''K''.


Algebraic extension

An element ''x'' of a field extension is algebraic over ''K'' if it is 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 su ...
of a nonzero
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 ''K''. For example, \sqrt 2 is algebraic over the rational numbers, because it is a root of x^2-2. If an element ''x'' of ''L'' is algebraic over ''K'', the
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\ ...
of lowest degree that has ''x'' as a root is called the minimal polynomial of ''x''. This minimal polynomial is irreducible over ''K''. An element ''s'' of ''L'' is algebraic over ''K'' if and only if the simple extension is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the ''K''-
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''K''(''s'') consists of 1, s, s^2, \ldots, s^, where ''d'' is the degree of the minimal polynomial. The set of the elements of ''L'' that are algebraic over ''K'' form a subextension, which is called the algebraic closure of ''K'' in ''L''. This results from the preceding characterization: if ''s'' and ''t'' are algebraic, the extensions and are finite. Thus is also finite, as well as the sub extensions , and (if ). It follows that , ''st'' and 1/''s'' are all algebraic. An ''algebraic extension'' is an extension such that every element of ''L'' is algebraic over ''K''. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example, \Q(\sqrt 2, \sqrt 3) is an algebraic extension of \Q, because \sqrt 2 and \sqrt 3 are algebraic over \Q. A simple extension 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 ...
it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic. Every field ''K'' has an algebraic closure, which is up to an isomorphism the largest extension field of ''K'' which is algebraic over ''K'', and also the smallest extension field such that every polynomial with coefficients in ''K'' has a root in it. For example, \Complex is an algebraic closure of \R, but not an algebraic closure of \Q, as it is not algebraic over \Q (for example is not algebraic over \Q).


Transcendental extension

:''See transcendence degree for examples and more extensive discussion of transcendental extensions.'' Given a field extension , a subset ''S'' of ''L'' is called algebraically independent over ''K'' if no non-trivial polynomial relation with coefficients in ''K'' exists among the elements of ''S''. The largest cardinality of an algebraically independent set is called the transcendence degree of ''L''/''K''. It is always possible to find a set ''S'', algebraically independent over ''K'', such that ''L''/''K''(''S'') is algebraic. Such a set ''S'' is called a
transcendence basis In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset o ...
of ''L''/''K''. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension ''L''/''K'' is said to be if and only if there exists a transcendence basis ''S'' of ''L''/''K'' such that ''L'' = ''K''(''S''). Such an extension has the property that all elements of ''L'' except those of ''K'' are transcendental over ''K'', but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form ''L''/''K'' where both ''L'' and ''K'' are algebraically closed. In addition, if ''L''/''K'' is purely transcendental and ''S'' is a transcendence basis of the extension, it doesn't necessarily follow that ''L'' = ''K''(''S''). For example, consider the extension \Q(x, \sqrt)/\Q, where ''x'' is transcendental over \Q. The set \ is algebraically independent since ''x'' is transcendental. Obviously, the extension \Q(x, \sqrt)/\Q(x) is algebraic, hence \ is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x for \sqrt. But it is easy to see that \ is a transcendence basis that generates \Q(x, \sqrt), so this extension is indeed purely transcendental.


Normal, separable and Galois extensions

An algebraic extension ''L''/''K'' is called normal if every irreducible polynomial in ''K'' 'X''that has a root in ''L'' completely factors into linear factors over ''L''. Every algebraic extension ''F''/''K'' admits a normal closure ''L'', which is an extension field of ''F'' such that ''L''/''K'' is normal and which is minimal with this property. An algebraic extension ''L''/''K'' is called separable if the minimal polynomial of every element of ''L'' over ''K'' is separable, i.e., has no repeated roots in an algebraic closure over ''K''. A Galois extension is a field extension that is both normal and separable. A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple). Given any field extension ''L''/''K'', we can consider its automorphism group Aut(''L''/''K''), consisting of all field automorphisms ''α'': ''L'' → ''L'' with ''α''(''x'') = ''x'' for all ''x'' in ''K''. When the extension is Galois this automorphism group is called the Galois group of the extension. Extensions whose Galois group is
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
are called abelian extensions. For a given field extension ''L''/''K'', one is often interested in the intermediate fields ''F'' (subfields of ''L'' that contain ''K''). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the intermediate fields and the subgroups of the Galois group, described by the
fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basi ...
.


Generalizations

Field extensions can be generalized to ring extensions which consist of a ring and one of its
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
s. A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be further generalized to Azumaya algebras, where the base field is replaced by a commutative
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
.


Extension of scalars

Given a field extension, one can " extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via complexification. In addition to vector spaces, one can perform extension of scalars for associative algebras defined over the field, such as polynomials or group algebras and the associated
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in extension of scalars: applications.


See also

* Field theory *
Glossary of field theory Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative rin ...
* Tower of fields * Primary extension * Regular extension


Notes


References

* * * *


External links

* {{springer, title=Extension of a field, id=p/e036970