Information about Extension Field
In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties.
Field extensions can be generalized to ring extension which consists of a ring and one of its subrings.
If L is an extension of F which is in turn an extension of K, then we say F is an intermediate field or subextension of the field extension L/K.
Given a field extension L/K and a subset S of L, we denote by K(S) the smallest subfield of L which contains K and S. We say K(S) is generated by the adjunction of elements of S to K. If S consists of only one element s we often write K(s) instead of K({s}). A field extension of the form L=K(s) is called a simple extension and s is called a primitive element of the extension.
Given a field extension L/K, then L can also be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars". We add the vectors just like we add elements in L, and scalar multiplication is multiplication of elements from L by elements from K. The dimension of this vector space is called the degree of the extension, and is denoted by [L : K].
An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension.
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 ring homomorphism between fields is injective, so field extensions are precisely the morphisms in the category of fields.
In the sequel, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q, also clearly a simple extension. The degree is 2 because {1, √2} can serve as a basis. Finite extensions of Q are also called algebraic number fields and are important in number theory.
Another extension field of the rationals, quite different in flavor, is the field of p-adic numbers Qp 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 K[X] in order to "create" a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 = -1. Then the polynomial X2 + 1 is irreducible in K[X], consequently the ideal (X2 + 1) generated by this polynomial is maximal, and L = K[X]/(X2 + 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 the 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 and n is a positive integer, we have a finite field GF(pn) with pn elements; this is an extension field of the finite field GF(p) = Z/pZ with p elements.
Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials 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 M, the set of all meromorphic functions defined on M is a field, denoted by C(M). It is an extension field of C, if we identify every complex number with the corresponding constant function defined on M.
Given an algebraic variety 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.
In particular then, the characteristics of L and K are the same.
If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it is said to be transcendental. If every element of L except those in K is transcendental over K, then the extension is said to be purely transcendental.
It can be shown that an extension is algebraic if and only if it is the union of its finite subextensions. In particular, every finite extension is algebraic. For example,
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. Given any algebraically independent set S over K, then K(S)/K is purely transcendental. 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 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 L. 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 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.
Field extensions can be generalized to ring extension which consists of a ring and one of its subrings.
Definitions
Given two fields K and L, if K is a subset of L and the field operations of addition and multiplication in K are the same as those in L, we say that K is a subfield of L, L is an extension field of K and that L/K, read as "L over K", is a field extension.If L is an extension of F which is in turn an extension of K, then we say F is an intermediate field or subextension of the field extension L/K.
Given a field extension L/K and a subset S of L, we denote by K(S) the smallest subfield of L which contains K and S. We say K(S) is generated by the adjunction of elements of S to K. If S consists of only one element s we often write K(s) instead of K({s}). A field extension of the form L=K(s) is called a simple extension and s is called a primitive element of the extension.
Given a field extension L/K, then L can also be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars". We add the vectors just like we add elements in L, and scalar multiplication is multiplication of elements from L by elements from K. The dimension of this vector space is called the degree of the extension, and is denoted by [L : K].
An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension.
Notes
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. 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 ring homomorphism between fields is injective, so field extensions are precisely the morphisms in the category of fields.
In the sequel, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
Examples
The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers Q. Clearly then, C/Q is also a field extension. We have [C : R] = 2 because {1,i} is a basis, so the extension C/R is finite. This is a simple extension because C=R(i). [R : Q] = c (the cardinality of the continuum), so this extension is infinite.The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q, also clearly a simple extension. The degree is 2 because {1, √2} can serve as a basis. Finite extensions of Q are also called algebraic number fields and are important in number theory.
Another extension field of the rationals, quite different in flavor, is the field of p-adic numbers Qp 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 K[X] in order to "create" a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 = -1. Then the polynomial X2 + 1 is irreducible in K[X], consequently the ideal (X2 + 1) generated by this polynomial is maximal, and L = K[X]/(X2 + 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 the 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 and n is a positive integer, we have a finite field GF(pn) with pn elements; this is an extension field of the finite field GF(p) = Z/pZ with p elements.
Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials 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 M, the set of all meromorphic functions defined on M is a field, denoted by C(M). It is an extension field of C, if we identify every complex number with the corresponding constant function defined on M.
Given an algebraic variety 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.
Elementary properties
If L/K is a field extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup of (L,+), and the multiplicative group (K-{0},·) is a subgroup of (L-{0},·). In particular, if x is an element of K, then its additive inverse −x computed in K is the same as the additive inverse of x computed in L; the same is true for multiplicative inverses of non-zero elements of K.In particular then, the characteristics of L and K are the same.
Algebraic and transcendental elements
If L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraic over K. Elements that are not algebraic are called transcendental. As an example:- In C/R, i is algebraic because it is a root of x2+1.
- In R/Q, √2 + √3 is algebraic, because it is a root of x4-10x2+1
- In R/Q, e is transcendental because there is no polynomial with rational coefficients that has e as a root (see transcendental number)
- In C/R, e is algebraic because it is the root of x-e
If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it is said to be transcendental. If every element of L except those in K is transcendental over K, then the extension is said to be purely transcendental.
It can be shown that an extension is algebraic if and only if it is the union of its finite subextensions. In particular, every finite extension is algebraic. For example,
- C/R and Q(√2)/Q, being finite, are algebraic.
- R/Q is transcendental, although not purely transcendental.
- K(X)/K is purely transcendental.
- R/Q is not simple, as it is neither finite nor purely transcendental.
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. Given any algebraically independent set S over K, then K(S)/K is purely transcendental. 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.
Normal, separable and Galois extensions
A field 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 L. 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 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.
See also
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.
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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.
Please refer to Glossary of field theory for some basic definitions in field theory.
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Please refer to Glossary of field theory for some basic definitions in field theory.
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, more specifically in ring theory, a ring extension or extension ring is a ring R with a subring S. We write R/S and say R is a ring extension of S
Given a ring extension R/
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Given a ring extension R/
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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In mathematics, a subring is a subset of a ring, which contains the multiplicative identity and is itself a ring under the same binary operations. Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity
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subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
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In mathematics, more specifically in abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.
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In mathematics, more specifically in 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.
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In mathematics, more specifically in 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.
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In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension.
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In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear
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In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra.
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quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element. The quotient group is written G/N and is usually spoken in English as G mod N
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non-injective function.]] In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y
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In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.
More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S
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More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S
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In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures.
The most common example occurs when the process is a function or map which preserves the structure in some sense.
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The most common example occurs when the process is a function or map which preserves the structure in some sense.
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In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion.
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction , where b is not zero.
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In mathematics, the cardinality of the continuum (sometimes also called the power of the continuum) is the cardinal number of the set of real numbers R (sometimes called the continuum). This cardinal number is often denoted by , so .
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In mathematics, an algebraic number field (or simply number field) F is a finite, (and hence algebraic) field extension of the field of rational numbers Q.
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Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.
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In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. For each prime number p, the p-adic number system extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number
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In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra.
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In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring.
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Definition of a polynomial
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root (or a zero) of a complex-valued function is a member of the domain of such that vanishes at , that is,
In other words, a "root" of a function is a value for that produces a result of zero ("0").
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In other words, a "root" of a function is a value for that produces a result of zero ("0").
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