In mathematics, a

polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...

''P''(''X'') over a given

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' is separable if its roots are distinct in an

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. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...

of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to

square-free polynomial In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if i ...

. If ''K'' is a

perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' i ...

then the two concepts coincide. In general, ''P''(''X'') is separable if and only if it is square-free over any field that contains ''K'', which holds if and only if ''P''(''X'') is

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...

to its

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 derivat ...

''D'' ''P''(''X'').

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.N. Jacobson, Basic Algebra I, p. 233 In this definition, separability depended on the field ''K''; for example, any polynomial over a

would have been considered separable. This definition, although it can be convenient for

Galois theory 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 ...

, is no longer in use.

Separable field extensions

Separable polynomials are used to define

separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomi ...

s: A

field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...

is a separable extension if and only if for every in which is algebraic over , the minimal polynomial of over is a separable polynomial.

Inseparable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomi ...

s (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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p'' is the characteristic. With this clue we can construct an example: :''P''(''X'') = ''X''^''p'' − ''T'' with ''K'' the field of

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s in the indeterminate ''T'' over the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) 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, subtr ...

with ''p'' elements. Here one can

prove Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

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 In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...

over the finite field, taking co-ordinates to their ''p''th power. Such mappings are fundamental to the

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...

of finite fields. Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory. (See

Radical morphism In algebraic geometry, a morphism of schemes :''f'': ''X'' → ''Y'' is called radicial or universally injective, if, for every field ''K'' the induced map ''X''(''K'') → ''Y''(''K'') is injective. (EGA I, (3.5.4)) This is a genera ...

for a higher-level discussion.) If ''L'' is the field extension :''K''(''T''^1/''p''), in other words the

splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polyn ...

of ''P'', then ''L''/''K'' is an example of a

purely inseparable field extension 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'', wit ...

. It is of degree ''p'', but has no

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 automorphis ...

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 In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...

of ''L'' with itself over ''K'' for this example has

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...

elements that are non-zero. This is another manifestation of inseparability: that is, the tensor product operation on fields need not produce a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

that is a product of fields (so, not a commutative

semisimple ring In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...

). If ''P''(''x'') is separable, and its roots form a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

of the field ''K''), then ''P''(''x'') is an

additive polynomial In mathematics, the additive polynomials are an important topic in classical algebraic number theory. Definition Let ''k'' be a field of prime characteristic ''p''. A polynomial ''P''(''x'') with coefficients in ''k'' is called an additive pol ...

Applications in Galois theory

Separable polynomials occur frequently in

. For example, let ''P'' be an irreducible polynomial with

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s 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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...

of the

Galois group 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 pol ...

of ''P''. Another example: ''P'' being as above, a resolvent ''R'' 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 Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...

root then the Galois group of ''P'' is contained in ''G''. For example, if ''D'' is the

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...

of ''P'' then

X^2-D

is a resolvent for the

alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...

. This resolvent is always separable (assuming the characteristic is not 2) if ''P'' is irreducible, but most resolvents are not always separable.

References

* Pages 240-241 of {{Lang Algebra, edition=3 Field (mathematics) Polynomials

Older definition

Separable field extensions

Applications in Galois theory

See also

References