Separable polynomial

Last updated May 20, 2024

In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial.^[1]

Older definition

In an older definition, P(X) was considered separable if each of its irreducible factors in K[X] is separable in the modern definition.^[2] In this definition, separability depended on the field K; for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use.^[3]

Separable field extensions

Separable polynomials are used to define separable extensions: A field extension $K \subset L$ is a separable extension if and only if for every $α$ in $L$ which is algebraic over $K$ , the minimal polynomial of $α$ over $K$ is a separable polynomial.

Inseparable extensions (that is, extensions which are not separable) may occur only in positive characteristic.

The criterion above leads to the quick conclusion that if P is irreducible and not separable, then D P(X) = 0. Thus we must have

P(X) = Q(X^p)

for some polynomial Q over K, where the prime number p is the characteristic.

With this clue we can construct an example:

P(X) = X^p − T

with K the field of rational functions in the indeterminate T over the finite field with p elements. Here one can prove directly that P(X) is irreducible and not separable. This is actually a typical example of why inseparability matters; in geometric terms P represents the mapping on the projective line over the finite field, taking co-ordinates to their pth power. Such mappings are fundamental to the algebraic geometry of finite fields. Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory. (See Radical morphism for a higher-level discussion.)

If L is the field extension

K(T^1/p),

in other words the splitting field of P, then L/K is an example of a purely inseparable field extension. It is of degree p, but has no automorphism fixing K, other than the identity, because T^1/p is the unique root of P. This shows directly that Galois theory must here break down. A field such that there are no such extensions is called perfect. That finite fields are perfect follows a posteriori from their known structure.

One can show that the tensor product of fields of L with itself over K for this example has nilpotent elements that are non-zero. This is another manifestation of inseparability: that is, the tensor product operation on fields need not produce a ring that is a product of fields (so, not a commutative semisimple ring).

If P(x) is separable, and its roots form a group (a subgroup of the field K), then P(x) is an additive polynomial.

Applications in Galois theory

Separable polynomials occur frequently in Galois theory.

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divide the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable (which is the case for every p but a finite number) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Another example: P being as above, a resolventR for a group G is a polynomial whose coefficients are polynomials in the coefficients of P, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then $X^{2}-D$ is a resolvent for the alternating group. This resolvent is always separable (assuming the characteristic is not 2) if P is irreducible, but most resolvents are not always separable.

Related Research Articles

In mathematics, an algebraic extension is a field extension $L / K$ such that every element of the larger field $L$ is algebraic over the smaller field $K$ ; that is, every element of $L$ is a root of a non-zero polynomial with coefficients in $K$ . A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.

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.

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod $p$ when $p$ is a prime number.

In mathematics, particularly in algebra, a field extension is a pair of fields $, such that the operations of K are those of L restricted to K . In this case, L is an extension field of K and K is a subfield of L . For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.$

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers $A$ is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial $x 2 - 2$ is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as $if it is considered as a polynomial with real coefficients. One says that the polynomial x 2 - 2 is irreducible over the integers but not over the reals.$

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.

In field theory, a branch of algebra, an algebraic field extension $is called a separable extension if for every, the minimal polynomial of over F is a separable polynomial. There is also a more general definition that applies when E is not necessarily algebraic over F . An extension that is not separable is said to be inseparable .$

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

In algebra, a field k is perfect if any one of the following equivalent conditions holds:

In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.

In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form x^q = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions.

In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.

In field theory, a branch of mathematics, the minimal polynomial of an element $α$ of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that $α$ is a root of the polynomial. If the minimal polynomial of $α$ exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1.

In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

References

↑ Pages 240-241 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
↑ N. Jacobson, Basic Algebra I, p. 233
↑ Sutherland, Andrew. "18.785 Number Theory I; Lecture 4: Étale algebras, norm, and trace" (PDF).

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Pages 240-241 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001

[2] N. Jacobson, Basic Algebra I, p. 233

[3] Sutherland, Andrew. "18.785 Number Theory I; Lecture 4: Étale algebras, norm, and trace" (PDF).

[1]

[2]

[3]