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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, particularly in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, a field extension is a pair of
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
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 Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
, the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s are an extension field of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
, and in the study of
polynomial roots In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
through
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, and are widely used in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
.


Subfield

A subfield of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
''L'' is a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
''K'' of ''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 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 add ...

inverse
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s is a subfield of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphic
to) a subfield of any field of
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
0. The
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. The
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 finitely generated over ''K''. If ''S'' consists of a single element ''s'', the extension is called a
simple extensionIn field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization ...
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 Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
0, every finite extension is a simple extension. This is the
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas an ...
, 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 fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s in ''s'' over ''K''.


Caveats

The notation ''L'' / ''K'' is purely formal and does not imply the formation of a
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
or
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factore ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

injective
ring homomorphism In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical an ...
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
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
s in the
category of fields Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience Experience refer ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...
), 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 field In mathematics, an algebraic number field (or simply number field) K is a finite Degree of a field extension, degree (and hence algebraic extension, algebraic) field extension of the field (mathematics), field of rational numbers Thus K is a fiel ...
s and are important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of
p-adic number In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real number, real and complex number systems. ...
s \Q_p for a prime number ''p''. It is common to construct an extension field of a given field ''K'' as a
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
of the
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''K'' 'X''in order to "create" a
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
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) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
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 In abstract algebra, a splitting field of a polynomial with coefficients in a field (mathematics), field is the smallest field extension of that field over which the polynomial ''splits'' or decomposes into linear factors. Definition A splitting f ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

rational function
s in the variable ''X'' with coefficients in ''K''; the elements of ''K''(''X'') are fractions of two
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s over ''K'', and indeed ''K''(''X'') is the
field of fractions In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''M'', the set of all
meromorphic function In the mathematical field of complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Anci ...
s 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

constant function
defined on ''M''. More generally, given an algebraic variety ''V'' over some field ''K'', then the function field of an algebraic variety, 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 plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
of a nonzero
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
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 of lowest degree that has ''x'' as a root is called the minimal polynomial (field theory), 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
''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 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

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 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 extension, 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 extension, separable if the minimal polynomial of every element of ''L'' over ''K'' is separable polynomial, 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 In field theory, the primitive element theorem is a result characterizing the finite degree field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas an ...
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 group, abelian 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 between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory.


Generalizations

Field extensions can be generalized to ring extensions which consist of a ring (mathematics), ring and one of its subrings. 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.


Extension of scalars

Given a field extension, one can "Extension of scalars, 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 ring, group algebras and the associated group representations. 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, extension of scalars: applications.


See also

* Field theory (mathematics), Field theory * Glossary of field theory * Tower of fields * Primary extension * Regular extension


Notes


References

* * * *


External links

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