Separable extension


In field theory, a subfield of algebra, a separable extension is an algebraic field extension such that for every, the minimal polynomial of over F is a separable polynomial. Otherwise, the extension 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.
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 supposed to be separable.
The extreme opposite of the concept of separable extension, namely the concept of purely inseparable extension, also occurs quite naturally, as every algebraic extension may be decomposed in a unique way as a purely inseparable extension of separable extension. An algebraic extension of fields of non-zero characteristics is a purely inseparable extension if and only if for every, the minimal polynomial of over is not a separable polynomial, or, equivalently, for every element of, there is a positive integer such that.

Informal discussion

An arbitrary polynomial with coefficients in some field is said to have distinct roots or to be square-free if it has roots in some extension field. For instance, the polynomial has precisely roots in the complex plane; namely and, and hence does have distinct roots. On the other hand, the polynomial, which is the square of a non-constant polynomial does not have distinct roots, as its degree is two, and 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 and reducible over some extension of. Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial over is divisible by a square over some field extension, then the greatest common divisor of and its derivative is not constant. Note that the coefficients of belong to the same field as those of, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of and has coefficients in. Since is irreducible in, this greatest common divisor is necessarily itself. Because the degree of is strictly less than the degree of, it follows that the derivative of is zero, which implies that the characteristic of the field is a prime number, and may be written
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 in is separable if and only if it has distinct roots in any extension of .
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 polynomial in is not separable, if and only if the characteristic of is a for some irreducible polynomial in. By repeated application of this property, it follows that in fact, for a non-negative integer and some separable irreducible polynomial in .
If the Frobenius endomorphism of is not surjective, there is an element which is not a th power of an element of. In this case, the polynomial is irreducible and inseparable. Conversely, if there exists an inseparable irreducible polynomial in, then the Frobenius endomorphism of cannot be an automorphism, since, otherwise, we would have for some, and the polynomial would factor as
If is a finite field of prime characteristic p, and if is an indeterminate, then the field of rational functions over,, is necessarily imperfect, and the polynomial is inseparable. More generally, if F is any field of 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 irreducible polynomials are separable. It follows that is perfect if and only if either has characteristic zero, or has prime characteristic and the Frobenius endomorphism of is an automorphism. This includes every finite field.

Separable elements and separable extensions

Let be a field extension. An element is separable over if it is algebraic over, and its minimal polynomial is separable.
If are separable over, then, and are separable over F.
Thus the set of all elements in separable over forms a subfield of, called the separable closure of in.
The separable closure of in an algebraic closure of is simply called the separable closure of. Like the algebraic closure, it is unique up to an isomorphism, and in general, this isomorphism is not unique.
A field extension is separable, if is the separable closure of in. This is the case if and only if is generated over by separable elements.
If are field extensions, then is separable over if and only if is separable over and is separable over.
If is a finite extension, then the following are equivalent.
  1. is separable over.
  2. where are separable elements of.
  3. where is a separable element of.
  4. If is an algebraic closure of, then there are exactly field homomorphisms of into which fix.
  5. For any normal extension of which contains, then there are exactly field homomorphisms of into which fix.
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 be an algebraic extension of fields of characteristic. The separable closure of in is For every element there exists a positive integer such that and thus is a purely inseparable extension of . It follows that is the unique intermediate field that is separable over and over which is purely inseparable.
If is a finite extension, its degree is the product of the degrees and . The former, often denoted is often referred to as the separable part of , or as the ' of ; the latter is referred to as the inseparable part of the degree or the '. The inseparable degree is 1 in characteristic zero and a power of in characteristic.
On the other hand, an arbitrary algebraic extension may not possess an intermediate extension that is purely inseparable over and over which is separable. However, such an intermediate extension may exist if, for example, is a finite degree normal extension. This equality implies that, if is finite, and is an intermediate field between and, then.
The separable closure of a field is the separable closure of in an algebraic closure of. It is the maximal Galois extension of. By definition, is perfect if and only if its separable and algebraic closures coincide.

Separability of transcendental extensions

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 is a transcendence basis of such that is a separable algebraic extension of. 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.
Let be a field extension of characteristic exponent . The following properties are equivalent:
where denotes the tensor product of fields, is the field of the th powers of the elements of , and is the field obtained by adjoining to the th root of all its elements.

Differential criteria

Separability can be studied with the aid of derivations. Let be a finitely generated field extension of a field. Denoting the -vector space of the -linear derivations of, one has
and the equality holds if and only if E is separable over F.
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, this matrix is invertible if and only if is a separating transcendence basis.