Editing Separable extension

{{Short description|Type of algebraic field extension}}
In [[field theory (mathematics)|field theory]], a branch of [[algebra]], an [[algebraic field extension]] <math>E/F</math> is called a '''separable extension''' if for every <math>\alpha\in E</math>, the [[minimal polynomial (field theory)|minimal polynomial]] of <math>\alpha</math> over {{mvar|F}} is a [[separable polynomial]] (i.e., its [[formal derivative]] is not the zero [[polynomial]], or equivalently it has no repeated [[zero of a function|root]]s in any extension field).<ref name="Isaacs281">Isaacs, p. 281</ref>  There is also a more general definition that applies when {{mvar|E}} is not necessarily algebraic over {{mvar|F}}.  An extension that is not separable is said to be ''inseparable''.

Every algebraic extension of a [[field (mathematics)|field]] of [[characteristic (algebra)#Case of fields|characteristic]] zero is separable, and every algebraic extension of a [[finite field]] is separable.<ref name="Isaacs18.11p281">Isaacs, Theorem 18.11, p. 281</ref>
It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the [[fundamental theorem of Galois theory]] is a theorem about [[normal extension]]s, which remains true in non-zero characteristic only if the extensions are also assumed to be separable.<ref>Isaacs, Theorem 18.13, p. 282</ref>

The opposite concept, a [[purely inseparable extension]], also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of  a separable extension. An algebraic extension <math>E/F</math> of fields of non-zero characteristic {{math|''p''}} is a purely inseparable extension if and only if for every <math>\alpha\in E\setminus F</math>, the minimal polynomial of <math>\alpha</math> over {{math|''F''}} is ''not'' a separable polynomial, or, equivalently, for every element {{math|''x''}} of {{math|''E''}}, there is a positive [[integer]] {{math|''k''}} such that <math>x^{p^k} \in F</math>.<ref name="Isaacs298">Isaacs, p. 298</ref>

The simplest nontrivial example of a (purely) inseparable extension is <math>E=\mathbb{F}_p(x) \supseteq F=\mathbb{F}_p(x^p)</math>, fields of [[rational function]]s in the indeterminate ''x'' with coefficients in the [[finite field]] <math>\mathbb{F}_p=\mathbb{Z}/(p)</math>. The element <math>x\in E</math> has minimal polynomial <math>f(X)=X^p -x^p \in F[X]</math>, having <math>f'(X) = 0</math> and a ''p''-fold multiple root, as <math>f(X)=(X-x)^p\in E[X]</math>. This is a [[simple field extension|simple]] algebraic extension of degree ''p'', as <math>E = F[x]</math>, but it is not a normal extension since the [[Galois group]] <math>\text{Gal}(E/F)</math> is [[trivial group|trivial]].

==Informal discussion==
An arbitrary polynomial {{math|''f''}} with coefficients in some field {{math|''F''}} is said to have ''distinct roots'' or to be [[square-free polynomial|square-free]] if it has {{math|deg ''f''}} roots in some [[extension field]] <math>E\supseteq F</math>. For instance, the polynomial {{math|1=''g''(''X'') = ''X''<sup>&thinsp;2</sup> − 1}} has precisely {{math|1=deg&thinsp;''g'' = 2}} roots in the [[complex plane]]; namely {{math|1}} and {{math|−1}}, and hence ''does have'' distinct roots. On the other hand, the polynomial {{math|1=''h''(''X'') = (''X'' − 2)<sup>2</sup>}}, which is the square of a non-constant polynomial ''does not'' have distinct roots, as its degree is two, and {{math|2}} is its only root.

Every polynomial may be factored in linear factors over an [[algebraic closure]] of the field of its coefficients. Therefore, the polynomial does not have distinct roots if and only if it is divisible by the square of a polynomial of positive degree. This is the case if and only if the [[polynomial greatest common divisor|greatest common divisor]] of the polynomial and its [[formal derivative|derivative]] is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots.

In this context, the case of irreducible polynomials requires some care. A priori, it may seem that being divisible by a square is impossible for an [[irreducible polynomial]], which has no non-constant divisor except itself. However, irreducibility depends on the ambient field, and a polynomial may be irreducible over {{math|''F''}} and reducible over some extension of {{math|''F''}}. Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial {{math|''f''}} over {{math|''F''}} is divisible by a square over some field extension, then (by the discussion above) the greatest common divisor of {{math|''f''}} and its derivative {{math|''f''{{′}}}} is not constant. Note that the coefficients of {{math|''f''{{′}}}} belong to the same field as those of {{math|''f''}}, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of {{math|''f''}} and {{math|''f''{{′}}}} has coefficients in {{math|''F''}}. Since {{math|''f''}} is irreducible in {{math|''F''}}, this greatest common divisor is necessarily {{math|''f''}} itself. Because the degree of {{math|''f''{{′}}}} is strictly less than the degree of {{math|''f''}}, it follows that the derivative of {{math|''f''}} is zero, which implies that the [[characteristic of a field|characteristic]] of the field is a prime number {{math|''p''}}, and {{math|''f''}} may be written
:<math>f(x)= \sum_{i=0}^ka_ix^{pi}.</math>

A polynomial such as this one, whose formal derivative is zero, is said to be ''inseparable''. Polynomials that are not inseparable are said to be ''separable''. A ''separable extension'' is an extension that may be generated by ''separable elements'', that is elements whose minimal polynomials are separable.

==Separable and inseparable polynomials==
An [[irreducible polynomial]] {{math|''f''}} in {{math|''F''[''X'']}} is [[separable polynomial|separable]] if and only if it has distinct roots in any [[field extension|extension]] of {{math|''F''}}. That is, if it is the product of distinct linear factors {{math|''X'' - ''a''}} in some [[algebraically closed field|algebraic closure]] of {{math|''F''}}.<ref>Isaacs, p. 280</ref> 
Let {{math|''f''}} in {{math|''F''[''X'']}} be an irreducible polynomial and {{math|''f'' '}} its [[formal derivative]]. Then the following are equivalent conditions for the irreducible polynomial {{math|''f''}} to be separable:
* If {{math|''E''}} is an extension of {{math|''F''}} in which {{math|''f''}} is a product of linear factors then no square of these factors divides {{math|''f''}} in {{math|''E''[''X'']}} (that is {{math|''f''}} is [[square-free polynomial|square-free]] over {{math|''E''}}).<ref name=IsaacsLem18.7>Isaacs, Lemma 18.7, p. 280</ref>
* There exists an extension {{math|''E''}} of {{math|''F''}} such that {{math|''f''}} has {{math|deg(''f'')}} pairwise distinct roots in {{math|''E''}}.<ref name=IsaacsLem18.7/>
* The constant {{math|1}} is a [[polynomial greatest common divisor]] of {{math|''f''}} and {{math|''f'' '}}.<ref>Isaacs, Theorem 19.4, p. 295</ref>
* The formal derivative {{math|''f'' '}} of {{math|''f''}} is not the zero polynomial.<ref>Isaacs, Corollary 19.5, p. 296</ref>
* Either the characteristic of  {{math|''F''}} is zero, or the characteristic is {{math|''p''}}, and {{math|''f''}} is not of the form <math>\textstyle\sum_{i=0}^k a_iX^{pi}.</math>

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 be separable, its coefficients must lie in a field of prime characteristic. More generally, an irreducible (non-zero) polynomial {{math|''f''}} in {{math|''F''[''X'']}} is not separable, if and only if the characteristic of {{math|''F''}} is a (non-zero) prime number {{math|''p''}}, and {{math|1=''f''(''X'')=''g''(''X''<sup>''p''</sup>}}) for some ''irreducible'' polynomial {{math|''g''}} in {{math|''F''[''X'']}}.<ref>Isaacs, Corollary 19.6, p. 296</ref> By repeated application of this property, it follows that in fact, <math>f(X)=g(X^{p^n})</math> for a non-negative integer {{math|''n''}} and some ''separable irreducible'' polynomial {{math|''g''}} in {{math|''F''[''X'']}} (where {{math|''F''}} is assumed to have prime characteristic ''p'').<ref>Isaacs, Corollary 19.9, p. 298</ref>

If the [[Frobenius endomorphism]] <math>x\mapsto x^p</math> of {{math|''F''}} is not surjective, there is an element <math>a\in F</math> that is not a {{math|''p''}}th power of an element of {{math|''F''}}. In this case, the polynomial <math>X^p-a</math> is irreducible and inseparable. Conversely, if there exists an inseparable irreducible (non-zero) polynomial <math>\textstyle f(X)=\sum a_iX^{ip}</math> in {{math|''F''[''X'']}}, then the [[Frobenius endomorphism]] of {{math|''F''}} cannot be an [[automorphism]], since, otherwise, we would have <math>a_i=b_i^p</math> for some <math>b_i</math>, and the polynomial {{math|''f''}} would factor as <math>\textstyle \sum a_iX^{ip}=\left(\sum b_iX^{i}\right)^p.</math><ref>Isaacs, Theorem 19.7, p. 297</ref>

If {{math|''K''}} is a finite field of prime characteristic ''p'', and if {{math|''X''}} is an [[indeterminate (variable)|indeterminate]], then the [[field of rational functions]] over {{math|''K''}}, {{math|''K''(''X'')}}, is necessarily [[Imperfect field|imperfect]], and the polynomial {{math|1=''f''(''Y'')=''Y''<sup>''p''</sup>−''X''}} is inseparable (its formal derivative in ''Y'' is 0).<ref name="Isaacs281"/> 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.<ref name="Isaacs299">Isaacs, p. 299</ref>

A field ''F'' is [[Perfect field|perfect]] if and only if all irreducible polynomials are separable. It follows that {{math|''F''}} is perfect if and only if either {{math|''F''}} has characteristic zero, or {{math|''F''}} has (non-zero) prime characteristic {{math|''p''}} and the [[Frobenius endomorphism]] of {{math|''F''}} is an automorphism. This includes every finite field.

==Separable elements and separable extensions==
Let <math>E\supseteq F</math> be a field extension. An element <math>\alpha\in E</math> is '''separable''' over {{math|''F''}} if it is algebraic over {{math|''F''}}, and its [[minimal polynomial (field theory)|minimal polynomial]] is separable (the minimal polynomial of an element is necessarily irreducible).

If <math>\alpha,\beta\in E</math> are separable over {{math|''F''}}, then <math>\alpha+\beta</math>, <math>\alpha\beta</math> and <math>1/\alpha</math> are separable over ''F''.

Thus the set of all elements in {{math|''E''}} separable over {{math|''F''}} forms a subfield of {{math|''E''}}, called the '''separable closure''' of {{math|''F''}} in {{math|''E''}}.<ref>Isaacs, Lemma 19.15, p. 300</ref>

The separable closure of {{math|''F''}} in an [[algebraic closure]] of {{math|''F''}} is simply called the '''[[separable closure]]''' of {{math|''F''}}. Like the algebraic closure, it is unique up to an isomorphism, and in general, this isomorphism is not unique.

A field extension <math>E\supseteq F</math> is '''separable''', if {{math|''E''}} is the separable closure of {{math|''F''}} in {{math|''E''}}. This is the case if and only if {{math|''E''}} is generated over {{math|''F''}} by separable elements.

If <math>E\supseteq L\supseteq F</math> are field extensions, then {{math|''E''}} is separable over {{math|''F''}} if and only if {{math|''E''}} is separable over {{math|''L''}} and {{math|''L''}} is separable over {{math|''F''}}.<ref>Isaacs, Corollary 18.12, p. 281 and Corollary 19.17, p. 301</ref>

If <math>E\supseteq F</math> is a [[finite extension]] (that is {{math|''E''}} is a {{math|''F''}}-[[vector space]] of finite [[dimension (vector space)|dimension]]), then the following are equivalent.
# {{math|''E''}} is separable over {{math|''F''}}.
# <math>E = F(a_1, \ldots, a_r)</math> where <math>a_1, \ldots, a_r</math> are separable elements of {{math|''E''}}.
# <math>E = F(a)</math> where {{math|''a''}} is a separable element of {{math|''E''}}.
# If {{math|''K''}} is an algebraic closure of {{math|''F''}}, then there are exactly <math>[E : F]</math> [[field homomorphism]]s of {{math|''E''}} into {{math|''K''}} that fix {{math|''F''}}.
# For any normal extension {{math|''K''}}  of {{math|''F''}} that contains {{math|''E''}}, then there are exactly <math>[E : F]</math> field homomorphisms of {{math|''E''}} into {{math|''K''}} that fix {{math|''F''}}.
The equivalence of 3. and 1. is known as the ''[[primitive element theorem]]'' or  ''Artin's theorem on primitive elements''.
Properties 4. and 5. are the basis of [[Galois theory]], and, in particular, of the [[fundamental theorem of Galois theory]].

==Separable extensions within algebraic extensions== 
Let <math>E \supseteq F</math> be an algebraic extension of fields of characteristic {{math|''p''}}. The separable closure of {{math|''F''}} in {{math|''E''}} is <math>S=\{\alpha\in E \mid \alpha \text{ is separable over } F\}.</math> For every element <math>x\in E\setminus S</math> there exists a positive integer {{math|''k''}} such that <math>x^{p^k}\in S,</math> and thus {{math|''E''}} is a [[purely inseparable extension]] of  {{math|''S''}}. It follows that  {{math|''S''}} is the unique intermediate field that is ''separable'' over {{math|''F''}} and over which {{math|''E''}} is ''purely inseparable''.<ref>Isaacs, Theorem 19.14, p. 300</ref>

If <math>E \supseteq F</math> is a [[finite extension]], its [[degree of a field extension|degree]]  {{math|[''E'' : ''F'']}} is the product of the degrees {{math|[''S'' : ''F'']}} and  {{math|[''E'' : ''S'']}}. The former, often denoted {{math|[''E'' : ''F'']<sub>sep</sub>}}, is referred to as the ''separable part'' of  {{math|[''E'' : ''F'']}},  or as the '''{{visible anchor|separable degree}}''' of {{math|''E''/''F''}}; the latter is referred to as the ''inseparable part'' of the degree or the '''{{visible anchor|inseparable degree}}'''.<ref name="Isaacs302">Isaacs, p. 302</ref> The inseparable degree is 1 in characteristic zero and a power of {{math|''p''}} in characteristic {{math|''p'' > 0}}.<ref>{{harvnb|Lang|2002|loc=Corollary V.6.2}}</ref>

On the other hand, an arbitrary algebraic extension <math>E\supseteq F</math> may not possess an intermediate extension {{math|''K''}} that is ''purely inseparable'' over {{math|''F''}} and over which {{math|''E''}} is ''separable''. However, such an intermediate extension may exist if, for example, <math>E\supseteq F</math> is a finite degree normal extension (in this case, {{math|''K''}} is the fixed field of the Galois group of {{math|''E''}} over {{math|''F''}}). Suppose that such an intermediate extension does exist, and {{math|[''E'' : ''F'']}} is finite, then {{math|1=[''S'' : ''F''] = [''E'' : ''K'']}}, where {{math|''S''}} is the separable closure of {{math|''F''}} in  {{math|''E''}}.<ref>Isaacs, Theorem 19.19, p. 302</ref> The known proofs of this equality use the fact that if <math>K\supseteq F</math> is a purely inseparable extension, and if {{math|''f''}} is a separable irreducible polynomial in {{math|''F''[''X'']}}, then {{math|''f''}} remains irreducible in ''K''[''X'']<ref>Isaacs, Lemma 19.20, p. 302</ref>). This equality implies that, if {{math|[''E'' : ''F'']}} is finite, and {{math|''U''}} is an intermediate field between {{math|''F''}} and {{math|''E''}}, then {{math|1=[''E'' : ''F'']<sub>sep</sub> = [''E'' : ''U'']<sub>sep</sub>⋅[''U'' : ''F'']<sub>sep</sub>}}.<ref>Isaacs, Corollary 19.21, p. 303</ref>

The separable closure {{math|''F''<sup>sep</sup>}} of a field {{math|''F''}} is the separable closure of {{math|''F''}} in an [[algebraic closure]] of {{math|''F''}}. It is the maximal [[Galois extension]] of {{math|''F''}}.  By definition, {{math|''F''}} is [[perfect field|perfect]] if and only if its separable and algebraic closures coincide.

== Separability of transcendental extensions ==

Separability problems may arise when dealing with [[transcendental extension]]s. This is typically the case for [[algebraic geometry]] over a field of prime characteristic, where the [[function field of an algebraic variety]] has a [[transcendence degree]] over the ground field that is equal to the [[dimension of an algebraic variety|dimension]] of the variety.

For defining the separability of a transcendental extension, it is natural to use the fact that every field extension is an algebraic extension of a [[purely transcendental extension]]. This leads to the following definition.

A ''separating transcendence basis'' of an extension <math>E\supseteq F</math> is a [[transcendence basis]]  {{math|''T''}} of {{math|''E''}} such that {{math|''E''}} is a separable algebraic extension of {{math|''F''(''T'')}}.  A [[finitely generated field extension]] is ''separable'' if and only it has a separating transcendence basis; an extension that is not finitely generated is called separable if every finitely generated subextension has a separating transcendence basis.<ref name=FJ38>Fried & Jarden (2008) p.38</ref>

Let <math>E\supseteq F</math> be a field extension of [[characteristic exponent of a field|characteristic exponent]] {{math|''p''}} (that is {{math|1=''p'' = 1}} in characteristic zero and, otherwise, {{math|''p''}} is the characteristic). The following properties are equivalent:
*{{math|''E''}} is a separable extension of {{math|''F''}},
*<math>E^p</math> and {{math|''F''}} are [[linearly disjoint]] over <math>F^p,</math>
*<math>F^{1/p} \otimes_F E</math> is [[reduced ring|reduced]],
*<math>L \otimes_F E</math> is reduced for every field extension {{math|''L''}} of {{math|''E''}},
where <math>\otimes_F</math> denotes the [[tensor product of fields]], <math>F^p</math> is the field of the {{math|''p''}}th powers of the elements of {{math|''F''}} (for any field {{math|''F''}}), and <math>F^{1/p}</math> is the field obtained by [[Adjunction (field theory)|adjoining]] to {{math|''F''}} the {{math|''p''}}th root of all its elements (see [[Separable algebra]] for details).

== Differential criteria ==
Separability can be studied with the aid of [[Kähler differential|derivation]]s. Let {{math|''E''}} be a [[finitely generated field extension]] of a field {{math|''F''}}. Denoting <math>\operatorname{Der}_F(E,E)</math> the {{math|''E''}}-vector space of the {{math|''F''}}-linear derivations of {{math|''E''}}, one has
:<math>\dim_E \operatorname{Der}_F(E,E) \ge \operatorname{tr.deg}_F E,</math>
and the equality holds if and only if ''E'' is separable over ''F'' (here "tr.deg" denotes the [[transcendence degree]]).

In particular, if <math>E/F</math> is an algebraic extension, then <math>\operatorname{Der}_F(E, E) = 0</math> if and only if <math>E/F</math> is separable.<ref name=FJ49>Fried & Jarden (2008) p.49</ref>

Let <math>D_1, \ldots, D_m</math> be a basis of <math>\operatorname{Der}_F(E,E)</math> and <math>a_1, \ldots, a_m \in E</math>. Then <math>E</math> is separable algebraic over <math>F(a_1, \ldots, a_m)</math> if and only if the matrix <math>D_i(a_j)</math> is invertible. In particular, when <math>m = \operatorname{tr.deg}_F E</math>, this matrix is invertible if and only if <math>\{ a_1, \ldots, a_m \}</math> is a separating transcendence basis.

==Notes==
{{Reflist|30em}}

==References==
* Borel, A. ''Linear algebraic groups'', 2nd ed.
* P.M. Cohn (2003). Basic algebra
* {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }}
* {{cite book
 | author = I. Martin Isaacs
 | author-link = Martin Isaacs
 | year = 1993
 | title = Algebra, a graduate course
 | edition = 1st
 | publisher = Brooks/Cole Publishing Company
 | isbn = 0-534-19002-2
}}
* {{cite book | first=Irving | last=Kaplansky | author-link=Irving Kaplansky | title=Fields and rings | edition=Second | series=Chicago lectures in mathematics | publisher=University of Chicago Press | year=1972 | isbn=0-226-42451-0 | zbl=1001.16500 | pages=55–59 }}
* {{Lang Algebra}}<!-- Lang 2000 Don't add a fullstop here: it breaks the layout! -->
* M. Nagata (1985). Commutative field theory: new edition, Shokabo. (Japanese) [http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1309-8.htm]
*{{cite book |last=Silverman |first=Joseph |title=The Arithmetic of Elliptic Curves |year=1993 |publisher=Springer |isbn=0-387-96203-4}}

==External links==
*{{Springer|id=s/s084470|title=separable extension of a field k}}

[[Category:Field extensions]]

[[de:Körpererweiterung#Separable Erweiterungen]]