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Separable extension

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Template:Short description In 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 of <math>\alpha</math> over Template:Mvar is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field).<ref name="Isaacs281">Isaacs, p. 281</ref> There is also a more general definition that applies when Template:Mvar is not necessarily algebraic over Template:Mvar. An extension that is not separable is said to be inseparable.

Every algebraic extension of a field of 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 extensions, 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 Template:Math 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 Template:Math is not a separable polynomial, or, equivalently, for every element Template:Math of Template:Math, there is a positive integer Template:Math 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 functions 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 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.

Informal discussionEdit

An arbitrary polynomial Template:Math with coefficients in some field Template:Math is said to have distinct roots or to be square-free if it has Template:Math roots in some extension field <math>E\supseteq F</math>. For instance, the polynomial Template:Math has precisely Template:Math roots in the complex plane; namely Template:Math and Template:Math, and hence does have distinct roots. On the other hand, the polynomial Template:Math, which is the square of a non-constant polynomial does not have distinct roots, as its degree is two, and Template:Math 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 greatest common divisor of the polynomial and its 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 Template:Math and reducible over some extension of Template:Math. Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial Template:Math over Template:Math is divisible by a square over some field extension, then (by the discussion above) the greatest common divisor of Template:Math and its derivative Template:Math is not constant. Note that the coefficients of Template:Math belong to the same field as those of Template:Math, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of Template:Math and Template:Math has coefficients in Template:Math. Since Template:Math is irreducible in Template:Math, this greatest common divisor is necessarily Template:Math itself. Because the degree of Template:Math is strictly less than the degree of Template:Math, it follows that the derivative of Template:Math is zero, which implies that the characteristic of the field is a prime number Template:Math, and Template:Math 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 polynomialsEdit

An irreducible polynomial Template:Math in Template:Math is separable if and only if it has distinct roots in any extension of Template:Math. That is, if it is the product of distinct linear factors Template:Math in some algebraic closure of Template:Math.<ref>Isaacs, p. 280</ref> Let Template:Math in Template:Math be an irreducible polynomial and Template:Math its formal derivative. Then the following are equivalent conditions for the irreducible polynomial Template:Math to be separable:

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 Template:Math in Template:Math is not separable, if and only if the characteristic of Template:Math is a (non-zero) prime number Template:Math, and Template:Math) for some irreducible polynomial Template:Math in Template:Math.<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 Template:Math and some separable irreducible polynomial Template:Math in Template:Math (where Template:Math 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 Template:Math is not surjective, there is an element <math>a\in F</math> that is not a Template:Mathth power of an element of Template:Math. 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 Template:Math, then the Frobenius endomorphism of Template:Math 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 Template:Math 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 Template:Math is a finite field of prime characteristic p, and if Template:Math is an indeterminate, then the field of rational functions over Template:Math, Template:Math, is necessarily imperfect, and the polynomial Template:Math 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 if and only if all irreducible polynomials are separable. It follows that Template:Math is perfect if and only if either Template:Math has characteristic zero, or Template:Math has (non-zero) prime characteristic Template:Math and the Frobenius endomorphism of Template:Math is an automorphism. This includes every finite field.

Separable elements and separable extensionsEdit

Let <math>E\supseteq F</math> be a field extension. An element <math>\alpha\in E</math> is separable over Template:Math if it is algebraic over Template:Math, and its minimal polynomial is separable (the minimal polynomial of an element is necessarily irreducible).

If <math>\alpha,\beta\in E</math> are separable over Template:Math, 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 Template:Math separable over Template:Math forms a subfield of Template:Math, called the separable closure of Template:Math in Template:Math.<ref>Isaacs, Lemma 19.15, p. 300</ref>

The separable closure of Template:Math in an algebraic closure of Template:Math is simply called the separable closure of Template:Math. 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 Template:Math is the separable closure of Template:Math in Template:Math. This is the case if and only if Template:Math is generated over Template:Math by separable elements.

If <math>E\supseteq L\supseteq F</math> are field extensions, then Template:Math is separable over Template:Math if and only if Template:Math is separable over Template:Math and Template:Math is separable over Template:Math.<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 Template:Math is a Template:Math-vector space of finite dimension), then the following are equivalent.

  1. Template:Math is separable over Template:Math.
  2. <math>E = F(a_1, \ldots, a_r)</math> where <math>a_1, \ldots, a_r</math> are separable elements of Template:Math.
  3. <math>E = F(a)</math> where Template:Math is a separable element of Template:Math.
  4. If Template:Math is an algebraic closure of Template:Math, then there are exactly <math>[E : F]</math> field homomorphisms of Template:Math into Template:Math that fix Template:Math.
  5. For any normal extension Template:Math of Template:Math that contains Template:Math, then there are exactly <math>[E : F]</math> field homomorphisms of Template:Math into Template:Math that fix Template:Math.

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 extensionsEdit

Let <math>E \supseteq F</math> be an algebraic extension of fields of characteristic Template:Math. The separable closure of Template:Math in Template:Math 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 Template:Math such that <math>x^{p^k}\in S,</math> and thus Template:Math is a purely inseparable extension of Template:Math. It follows that Template:Math is the unique intermediate field that is separable over Template:Math and over which Template:Math is purely inseparable.<ref>Isaacs, Theorem 19.14, p. 300</ref>

If <math>E \supseteq F</math> is a finite extension, its degree Template:Math is the product of the degrees Template:Math and Template:Math. The former, often denoted Template:Math, is referred to as the separable part of Template:Math, or as the Template:Visible anchor of Template:Math; the latter is referred to as the inseparable part of the degree or the Template:Visible anchor.<ref name="Isaacs302">Isaacs, p. 302</ref> The inseparable degree is 1 in characteristic zero and a power of Template:Math in characteristic Template:Math.<ref>Template:Harvnb</ref>

On the other hand, an arbitrary algebraic extension <math>E\supseteq F</math> may not possess an intermediate extension Template:Math that is purely inseparable over Template:Math and over which Template:Math 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, Template:Math is the fixed field of the Galois group of Template:Math over Template:Math). Suppose that such an intermediate extension does exist, and Template:Math is finite, then Template:Math, where Template:Math is the separable closure of Template:Math in Template:Math.<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 Template:Math is a separable irreducible polynomial in Template:Math, then Template:Math remains irreducible in K[X]<ref>Isaacs, Lemma 19.20, p. 302</ref>). This equality implies that, if Template:Math is finite, and Template:Math is an intermediate field between Template:Math and Template:Math, then Template:Math.<ref>Isaacs, Corollary 19.21, p. 303</ref>

The separable closure Template:Math of a field Template:Math is the separable closure of Template:Math in an algebraic closure of Template:Math. It is the maximal Galois extension of Template:Math. By definition, Template:Math is perfect if and only if its separable and algebraic closures coincide.

Separability of transcendental extensionsEdit

Separability problems may arise when dealing with transcendental extensions. 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 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 Template:Math of Template:Math such that Template:Math is a separable algebraic extension of Template:Math. 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 Template:Math (that is Template:Math in characteristic zero and, otherwise, Template:Math is the characteristic). The following properties are equivalent:

where <math>\otimes_F</math> denotes the tensor product of fields, <math>F^p</math> is the field of the Template:Mathth powers of the elements of Template:Math (for any field Template:Math), and <math>F^{1/p}</math> is the field obtained by adjoining to Template:Math the Template:Mathth root of all its elements (see Separable algebra for details).

Differential criteriaEdit

Separability can be studied with the aid of derivations. Let Template:Math be a finitely generated field extension of a field Template:Math. Denoting <math>\operatorname{Der}_F(E,E)</math> the Template:Math-vector space of the Template:Math-linear derivations of Template:Math, 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.

NotesEdit

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ReferencesEdit

External linksEdit

de:Körpererweiterung#Separable Erweiterungen

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