Separable extension
Encyclopedia
In modern algebra, an algebraic field extension 
is a separable extension if and only if for every 
, the minimal polynomial
of
over F is a separable polynomial
(i.e., has distinct roots). Otherwise, the extension is called inseparable. There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article.
The class of separable extensions is an extremely important one due to the fundamental role it plays in Galois theory
. More specifically, a finite degree field extension is Galois
if and only if it is both normal
and separable. Since algebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstacle in most applications of Galois theory
. For instance, every algebraic (in particular, finite degree) extension of the field of rational numbers is necessarily separable.
Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class of purely inseparable extensions, also occurs quite naturally. An algebraic extension
is a purely inseparable extension if and only if for every 
, the minimal polynomial of 
over F is not a separable polynomial
(i.e., does not have distinct roots). For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect field
is necessarily separable.
The study of separable extensions in their own right has far-reaching consequences. For instance, consider the result: "If E is a field with the property that every nonconstant polynomial with coefficients in E has a root in E, then E is algebraically closed." Despite its simplicity, it suggests a deeper conjecture: "If
is an algebraic extension and if every nonconstant polynomial with coefficients in F has a root in E, is E algebraically closed?" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions; for instance, in the case corresponding to the extension 
being separable, one known proof involves the use of the primitive element theorem
in the context of Galois extension
s.
An arbitrary polynomial f with coefficients in some field F is said to have distinct roots if and only if it has deg(f) roots in some extension field
. For instance, the polynomial g(X)=X2+1 with real coefficients has precisely deg(g)=2 roots in the complex plane; namely the imaginary unit
i, and its additive inverse −i, and hence does have distinct roots. On the other hand, the polynomial h(X)=(X−2)2 with real coefficients does not have distinct roots; only 2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(f)=2 roots.
In general, it can be shown that the polynomial f with coefficients in F has distinct roots if and only if for any extension field
, and any 
, 
does not divide f in E[X]. For instance, in the above paragraph, one observes that g has distinct roots and indeed g(X)=(X+i)(X−i) in the complex plane (and hence cannot have any factor of the form 
for any 
in the complex plane). On the other hand, h does not have distinct roots and indeed, h(X)=(X−2)2 in the complex plane (and hence does have a factor of the form 
for 
).
Although an arbitrary polynomial with rational or real coefficients may not have distinct roots, it is natural to ask at this stage whether or not there exists an irreducible polynomial
with rational or real coefficients that does not have distinct roots. The polynomial h(X)=(X−2)2 does not have distinct roots but it is not irreducible as it has a non-trivial factor (X−2). In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does not have distinct roots; in the language of field theory, every algebraic extension
of
or 
is separable and hence both of these fields are perfect
.
, E, for g over F, g is necessarily separable in E[X] since an arbitrary irreducible factor of g in E[X] is linear and hence has distinct roots. Despite this, a separable polynomial h in F[X] must necessarily be separable over every extension field of F.
Let f in F[X] be an irreducible polynomial and f' its formal derivative
. Then the following are equivalent conditions for f to be separable; that is, to have distinct roots:
By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero. Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not have distinct roots its coefficients must lie in a field of prime characteristic. More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must the characteristic of F be a (non-zero) prime number p, but also f(X)=g(Xp) for some irreducible polynomial g in F[X]. By repeated application of this property, it follows that in fact,
for a non-negative integer n and some separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).
By the property noted in the above paragraph, if f is an irreducible (non-zero) polynomial with coefficients in the field F of prime characteristic p, and does not have distinct roots, it is possible to write f(X)=g(Xp). Furthermore, if
, and if the Frobenius endomorphism
of F is an automorphism
, g may be written as
, and in particular, 
; a contradiction of the irreducibility of f. Therefore, if F[X] possesses an inseparable irreducible (non-zero) polynomial, then the Frobenius endomorphism of F cannot be an automorphism (where F is assumed to have prime characteristic p).
If K is a finite field of prime characteristic p, and if X is an indeterminant, then the field of rational functions over K, K(X), is necessarily imperfect. Furthermore, the polynomial f(Y)=Yp−X is inseparable. (To see this, note that there is some extension field
in which f has a root 
; necessarily, 
in E. Therefore, working over E, 
(the final equality in the sequence follows from freshman's dream
), and f does not have distinct roots.) More generally, if F is any field of (non-zero) prime characteristic for which the Frobenius endomorphism
is not an automorphism, F possesses an inseparable algebraic extension.
A field F is perfect
if and only if all of its algebraic extensions are separable (in fact, all algebraic extensions of F are separable if and only if all finite degree extensions of F are separable). By the argument outlined in the above paragraphs, it follows that F is perfect if and only if F has characteristic zero, or F has (non-zero) prime characteristic p and the Frobenius endomorphism
of F is an automorphism.
is a purely inseparable extension if and only if for every 
, the minimal polynomial of 
over F is not a separable polynomial
. If F is any field, the trivial extension
is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.
Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If
is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:
1. E is purely inseparable over F
2. For each element
, there exists 
such that 
.
3. Each element of E has minimal polynomial over F of the form
for some integer 
and some element 
.
It follows from the above equivalent characterizations that if
(for F a field of prime characteristic) such that 
for some integer 
, then E is purely inseparable over F. (To see this, note that the set of all x such that 
for some 
forms a field; since this field contains both 
and F, it must be E, and by condition 2 above, 
must be purely inseparable.)
If F is an imperfect field of prime characteristic p, choose
such that a is not a pth power in F, and let f(X)=Xp−a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose 
with 
. In particular, 
and by the property stated in the paragraph directly above, it follows that 
is a non-trivial purely inseparable extension (in fact, 
, and so 
is automatically a purely inseparable extension).
Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry
over fields of prime characteristic. If K is a field of characteristic p, and if V is an algebraic variety
over K of dimension greater than zero, the function field
K(V) is a purely inseparable extension over the subfield K(V)p of pth powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by p on an elliptic curve
over a finite field of characteristic p.
is an algebraic extension and if 
, then S is the unique intermediate field that is separable over F and over which E is purely inseparable. If 
is a finite degree extension, the degree [S : F] is referred to as the separable part of the degree of the extension 
(or the separable degree of E/F), and is often denoted by [E : F]sep or [E : F]s. The inseparable degree of E/F is the quotient of the degree by the separable degree. When the characteristic of F is p > 0, it is a power of p. Since the extension 
is separable if and only if 
, it follows that for separable extensions, [E : F]=[E : F]sep, and conversly. If 
is not separable (i.e., inseparable), then [E : F]sep is necessarily a non-trivial divisor of [E : F], and the quotient is necessarily a power of the characteristic of F.
On the other hand, an arbitrary algebraic extension
may not possess an intermediate extension K that is purely inseparable over F and over which E is separable (however, such an intermediate extension does exist when 
is a finite degree normal extension (in this case, K can be the fixed field of the Galois group of E over F)). If such an intermediate extension does exist, and if [E : F] is finite, then if S is defined as in the previous paragraph, [E : F]sep=[S : F]=[E : K]. One known proof of this result depends on the primitive element theorem
, but there does exist a proof of this result independent of the primitive element theorem (both proofs use the fact that if
is a purely inseparable extension, and if f in F[X] is a separable irreducible polynomial, then f remains irreducible in K[X]). The equality above ([E : F]sep=[S : F]=[E : K]) may be used to prove that if 
is such that [E : F] is finite, then [E : F]sep=[E : U]sep[U : F]sep.
If F is any field, the separable closure Fsep of F is the field of all elements in the algebraic closure
of F that are separable over F. The separable closure of F is useful in that Galois theory
may be carried out within it. Of course, if F is perfect, then the separable closure of F is precisely its algebraic closure (in particular, the notion of a separable closure is only interesting in the context of imperfect fields).
Let
be a field extension and let p be the characteristic exponent of 
. For any field extension L of k, we write 
(cf. Tensor product of fields
.) Then F is said to be separable over
if the following equivalent conditions are met:
(In other words, F is separable over k if F is a separable k-algebra
.)
Suppose there is some field extension L of k such that
is a domain. Then 
is separable over k if and only if the field of fractions of 
is separable over L.
An algebraic element of F is said to be separable over
if its minimal polynomial is separable. If 
is an algebraic extension, then the following are equivalent.
If
is finite extension, then the following are equivalent.
In the above, (iii) is known as the primitive element theorem
.
Fix the algebraic closure
, and denote by 
the set of all elements of 
that are separable over k. 
is then separable algebraic over k and any separable algebraic subextension of 
is contaiend in 
; it is called the separable closure of k (inside 
). 
is then purely inseparable over 
. Put in another way, k is perfect if and only if 
.
s. Let
be a finitely generated field extension of a field 
. Then
where the equality holds if and only if F is separable over k.
In particular, if
is an algebraic extension, then 
if and only if 
is separable.
Let
be a basis of 
and 
. Then 
is separable algebraic over 
if and only if the matrix 
is invertible. In particular, when 
, 
above is called the separating transcendence basis.
is a separable extension if and only if for every , the minimal polynomialMinimal polynomial (field theory)
In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...
of
over F is a separable polynomialSeparable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
(i.e., has distinct roots). Otherwise, the extension is called inseparable. There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article.
The class of separable extensions is an extremely important one due to the fundamental role it plays in Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
. More specifically, a finite degree field extension is Galois
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
if and only if it is both normal
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
and separable. Since algebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstacle in most applications of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
. For instance, every algebraic (in particular, finite degree) extension of the field of rational numbers is necessarily separable.
Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class of purely inseparable extensions, also occurs quite naturally. An algebraic extension
is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomialSeparable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
(i.e., does not have distinct roots). For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect field
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
is necessarily separable.
The study of separable extensions in their own right has far-reaching consequences. For instance, consider the result: "If E is a field with the property that every nonconstant polynomial with coefficients in E has a root in E, then E is algebraically closed." Despite its simplicity, it suggests a deeper conjecture: "If
is an algebraic extension and if every nonconstant polynomial with coefficients in F has a root in E, is E algebraically closed?" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theoremPrimitive element theorem
In mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
in the context of Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
s.
Informal discussion
The reader may wish to assume that, in what follows, F is the field of rational, real or complex numbers, unless otherwise stated.An arbitrary polynomial f with coefficients in some field F is said to have distinct roots if and only if it has deg(f) roots in some extension field
. For instance, the polynomial g(X)=X2+1 with real coefficients has precisely deg(g)=2 roots in the complex plane; namely the imaginary unitImaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
i, and its additive inverse −i, and hence does have distinct roots. On the other hand, the polynomial h(X)=(X−2)2 with real coefficients does not have distinct roots; only 2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(f)=2 roots.
In general, it can be shown that the polynomial f with coefficients in F has distinct roots if and only if for any extension field
, and any , does not divide f in E[X]. For instance, in the above paragraph, one observes that g has distinct roots and indeed g(X)=(X+i)(X−i) in the complex plane (and hence cannot have any factor of the form for any in the complex plane). On the other hand, h does not have distinct roots and indeed, h(X)=(X−2)2 in the complex plane (and hence does have a factor of the form for ).Although an arbitrary polynomial with rational or real coefficients may not have distinct roots, it is natural to ask at this stage whether or not there exists an irreducible polynomial
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
with rational or real coefficients that does not have distinct roots. The polynomial h(X)=(X−2)2 does not have distinct roots but it is not irreducible as it has a non-trivial factor (X−2). In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does not have distinct roots; in the language of field theory, every algebraic extension
Algebraic 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...
of
or is separable and hence both of these fields are perfectPerfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
.
Separable and inseparable polynomials
A polynomial f in F[X] is a separable polynomial if and only if every irreducible factor of f in F[X] has distinct roots. The separability of a polynomial depends on the field in which its coefficients are considered to lie; for instance, if g is an inseparable polynomial in F[X], and one considers a splitting fieldSplitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
, E, for g over F, g is necessarily separable in E[X] since an arbitrary irreducible factor of g in E[X] is linear and hence has distinct roots. Despite this, a separable polynomial h in F[X] must necessarily be separable over every extension field of F.
Let f in F[X] be an irreducible polynomial and f
Formal derivative
In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a...
. Then the following are equivalent conditions for f to be separable; that is, to have distinct roots:
- If
and 
, then 
does not divide f in E[X]. - There exsits
such that f has deg(f) roots in K. - f and f
' do not have a common root in any extension field of F. - f
' is not the zero polynomial.
By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero. Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not have distinct roots its coefficients must lie in a field of prime characteristic. More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must the characteristic of F be a (non-zero) prime number p, but also f(X)=g(Xp) for some irreducible polynomial g in F[X]. By repeated application of this property, it follows that in fact,
for a non-negative integer n and some separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).By the property noted in the above paragraph, if f is an irreducible (non-zero) polynomial with coefficients in the field F of prime characteristic p, and does not have distinct roots, it is possible to write f(X)=g(Xp). Furthermore, if
, and if the Frobenius endomorphismFrobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
of F is an automorphism
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 automorphisms of an object forms a group, called the automorphism...
, g may be written as
, and in particular, ; a contradiction of the irreducibility of f. Therefore, if F[X] possesses an inseparable irreducible (non-zero) polynomial, then the Frobenius endomorphism of F cannot be an automorphism (where F is assumed to have prime characteristic p).If K is a finite field of prime characteristic p, and if X is an indeterminant, then the field of rational functions over K, K(X), is necessarily imperfect. Furthermore, the polynomial f(Y)=Yp−X is inseparable. (To see this, note that there is some extension field
in which f has a root ; necessarily, in E. Therefore, working over E, (the final equality in the sequence follows from freshman's dreamFreshman's dream
The freshman's dream is a name sometimes given to the error n = xn + yn, where n is a real number . Beginning students commonly make this error in computing the exponential of a sum of real numbers...
), and f does not have distinct roots.) More generally, if F is any field of (non-zero) prime characteristic for which the Frobenius endomorphism
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
is not an automorphism, F possesses an inseparable algebraic extension.
A field F is perfect
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
if and only if all of its algebraic extensions are separable (in fact, all algebraic extensions of F are separable if and only if all finite degree extensions of F are separable). By the argument outlined in the above paragraphs, it follows that F is perfect if and only if F has characteristic zero, or F has (non-zero) prime characteristic p and the Frobenius endomorphism
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
of F is an automorphism.
Properties
- If
is an algebraic field extension, and if 
are separable over F, then 
and 
are separable over F. In particular, the set of all elements in E separable over F forms a field. - If
is such that 
and 
are separable extensions, then 
is separable. Conversely, if 
is a separable algebraic extension, and if L is any intermediate field, then 
and 
are separable extensions. - If
is a finite degree separable extension, then it has a primitive element; i.e., there exists 
with 
. This fact is also known as the primitive element theoremPrimitive element theoremIn mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
or Artin's theorem on primitive elements.
Purely inseparable extensions
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomialSeparable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
. If F is any field, the trivial extension
is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If
is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:1. E is purely inseparable over F
2. For each element
, there exists such that .3. Each element of E has minimal polynomial over F of the form
for some integer and some element .It follows from the above equivalent characterizations that if
(for F a field of prime characteristic) such that for some integer , then E is purely inseparable over F. (To see this, note that the set of all x such that for some forms a field; since this field contains both and F, it must be E, and by condition 2 above, must be purely inseparable.)If F is an imperfect field of prime characteristic p, choose
such that a is not a pth power in F, and let f(X)=Xp−a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose with . In particular, and by the property stated in the paragraph directly above, it follows that is a non-trivial purely inseparable extension (in fact, , and so is automatically a purely inseparable extension).Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
over fields of prime characteristic. If K is a field of characteristic p, and if V is an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
over K of dimension greater than zero, the function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
K(V) is a purely inseparable extension over the subfield K(V)p of pth powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by p on an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
over a finite field of characteristic p.
Properties
- If the characteristic of a field F is a (non-zero) prime number p, and if
is a purely inseparable extension, then if 
, K is purely inseparable over F and E is purely inseparable over K. Furthermore, if [E : F] is finite, then it is a power of p, the characteristic of F. - Conversely, if
is such that 
and 
are purely inseparable extensions, then E is purely inseparable over F. - An algebraic extension
is an inseparable extension if and only if there is some 
such that the minimal polynomial of 
over F is not a separable polynomialSeparable polynomialIn mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
(i.e., an algebraic extension is inseparable if and only if it is not separable; note, however, that an inseparable extension is not the same thing as a purely inseparable extension). If
is a finite degree non-trivial inseparable extension, then [E : F] is necessarily divisible by the characteristic of F. - If
is a finite degree normal extension, and if 
, then K is purely inseparable over F and E is separable over K.
Separable extensions within algebraic extensions
Separable extensions occur quite naturally within arbitrary algebraic field extensions. More specifically, if is an algebraic extension and if , then S is the unique intermediate field that is separable over F and over which E is purely inseparable. If is a finite degree extension, the degree [S : F] is referred to as the separable part of the degree of the extension (or the separable degree of E/F), and is often denoted by [E : F]sep or [E : F]s. The inseparable degree of E/F is the quotient of the degree by the separable degree. When the characteristic of F is p > 0, it is a power of p. Since the extension is separable if and only if , it follows that for separable extensions, [E : F]=[E : F]sep, and conversly. If is not separable (i.e., inseparable), then [E : F]sep is necessarily a non-trivial divisor of [E : F], and the quotient is necessarily a power of the characteristic of F.On the other hand, an arbitrary algebraic extension
may not possess an intermediate extension K that is purely inseparable over F and over which E is separable (however, such an intermediate extension does exist when is a finite degree normal extension (in this case, K can be the fixed field of the Galois group of E over F)). If such an intermediate extension does exist, and if [E : F] is finite, then if S is defined as in the previous paragraph, [E : F]sep=[S : F]=[E : K]. One known proof of this result depends on the primitive element theoremPrimitive element theorem
In mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
, but there does exist a proof of this result independent of the primitive element theorem (both proofs use the fact that if
is a purely inseparable extension, and if f in F[X] is a separable irreducible polynomial, then f remains irreducible in K[X]). The equality above ([E : F]sep=[S : F]=[E : K]) may be used to prove that if is such that [E : F] is finite, then [E : F]sep=[E : U]sep[U : F]sep.If F is any field, the separable closure Fsep of F is the field of all elements in the algebraic closure
Algebraic 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 F that are separable over F. The separable closure of F is useful in that Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
may be carried out within it. Of course, if F is perfect, then the separable closure of F is precisely its algebraic closure (in particular, the notion of a separable closure is only interesting in the context of imperfect fields).
The definition of separable non-algebraic extension fields
Although many important applications of the theory of separable extensions stem from the context of algebraic field extensions, there are important instances in mathematics where it is profitable to study (not necessarily algebraic) separable field extensions.Let
be a field extension and let p be the characteristic exponent of . For any field extension L of k, we write (cf. Tensor product of fieldsTensor product of fields
In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K...
.) Then F is said to be separable over
if the following equivalent conditions are met:
and 
are linearly disjointLinearly disjointIn mathematics, algebras A, B over a field k inside some field extension \Omega of k are said to be linearly disjoint over k if the following equivalent conditions are met:...
over
is reduced.
is reduced for all field extensions L of k.
(In other words, F is separable over k if F is a separable k-algebra
Separable algebra
A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.- Definition and First Properties :Let K be a field...
.)
Suppose there is some field extension L of k such that
is a domain. Then is separable over k if and only if the field of fractions of is separable over L.An algebraic element of F is said to be separable over
if its minimal polynomial is separable. If is an algebraic extension, then the following are equivalent.
- F is separable over k.
- F consists of elements that are separable over k.
- Every subextension of F/k is separable.
- Every finite subextension of F/k is separable.
If
is finite extension, then the following are equivalent.
- (i) F is separable over k.
- (ii)
where 
are separable over k. - (iii) In (ii), one can take
- (iv) For some very large field
, there are precisely 
k-isomorphisms from 
to 
.
In the above, (iii) is known as the primitive element theorem
Primitive element theorem
In mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
.
Fix the algebraic closure
, and denote by the set of all elements of that are separable over k. is then separable algebraic over k and any separable algebraic subextension of is contaiend in ; it is called the separable closure of k (inside ). is then purely inseparable over . Put in another way, k is perfect if and only if .Differential criteria
The separability can be studied with the aid of derivations and Kähler differentialKähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
s. Let
be a finitely generated field extension of a field . Thenwhere the equality holds if and only if F is separable over k.
In particular, if
is an algebraic extension, then if and only if is separable.Let
be a basis of and . Then is separable algebraic over if and only if the matrix is invertible. In particular, when , above is called the separating transcendence basis.See also
- Separable polynomialSeparable polynomialIn mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
- Perfect fieldPerfect fieldIn algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
- Primitive element theoremPrimitive element theoremIn mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
- Normal extensionNormal extensionIn abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
- Galois extensionGalois extensionIn mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
- Algebraic closureAlgebraic closureIn 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....
- Derivation