Editing Separable polynomial

{{Short description|Polynomial coprime with its derivative}}
In [[mathematics]], a [[polynomial]] ''P''(''X'') over a given [[field (mathematics)|field]] ''K'' is '''separable''' if its [[root of a polynomial|roots]] are [[distinct (mathematics)|distinct]] in an [[algebraic closure]] of ''K'', that is, the number of distinct roots is equal to the [[degree of a polynomial|degree]] of the polynomial.<ref>Pages 240-241 of {{Lang Algebra|edition=3}}</ref>

This concept is closely related to [[square-free polynomial]]. If ''K'' is a [[perfect field]] 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 polynomials|coprime]] to its [[formal derivative]] ''D''&thinsp;''P''(''X'').

==Older definition==
In an older definition, ''P''(''X'') was considered separable if each of its [[irreducible polynomial|irreducible]] factors in ''K''[''X''] is separable in the modern definition.<ref>N. Jacobson, Basic Algebra I, p. 233</ref> 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.<ref>{{cite web|last=Sutherland|first=Andrew|author-link=Andrew Sutherland (mathematician)|title=18.785 Number Theory I; Lecture 4: Étale algebras, norm, and trace|url=https://math.mit.edu/classes/18.785/2019fa/LectureNotes4.pdf}}</ref>

==Separable field extensions==
Separable polynomials are used to define [[separable extension]]s: A [[field extension]] {{math|''K'' ⊂ ''L''}} is a separable extension if and only if for every {{math|''α''}} in {{mvar|L}} which is [[algebraic element|algebraic]] over {{mvar|K}}, the [[minimal polynomial (field theory)|minimal polynomial]] of  {{math|''α''}} over {{mvar|K}} is a separable polynomial.

[[Inseparable extension]]s (that is, extensions which are not separable) may occur only in positive [[characteristic (algebra)|characteristic]]. 

The criterion above leads to the quick conclusion that if ''P'' is irreducible and not separable, then ''D''&thinsp;''P''(''X'') = 0.
Thus we must have
:''P''(''X'') = ''Q''(''X''<sup>''p''</sup>) 
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''<sup>''p''</sup> − ''T'' 
with ''K'' the field of [[rational function]]s in the indeterminate ''T'' over the [[finite field]] with ''p'' elements. Here one can [[mathematical proof|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 ''p''th 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''<sup>1/''p''</sup>), 
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''<sup>1/''p''</sup> 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 (mathematics)|ring]] that is a product of fields (so, not a [[commutative ring|commutative]] [[semisimple ring]]).

If ''P''(''x'') is separable, and its roots form a [[group (mathematics)|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]] [[coefficient]]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 [[modular arithmetic|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 [[cyclic permutation|cycles]] of some [[permutation]] of the [[Galois group]] 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 number|rational]] root then the Galois group of ''P'' is contained in ''G''. For example, if ''D'' is the [[discriminant]] of ''P'' then <math>X^2-D</math> 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.

==See also==
*[[Frobenius endomorphism]]

==References==
{{Reflist}}

[[Category:Field (mathematics)]]
[[Category:Polynomials]]