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In [[mathematics]], two slightly different notions of '''separable polynomial''' are used, by different authors.
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According to the most common one, a [[polynomial]] ''P''(''X'') over a given [[field (mathematics)|field]] ''K'' is '''separable''' if its roots are [[distinct]] in an [[algebraic closure]] of ''K'', that is the number of its distinct roots is equal to its degree.<ref>S. Lang, Algebra, p. 178</ref> In the context of [[polynomial factorization]], such a separable polynomial is also called a [[square-free polynomial]].
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For the second definition, ''P''(''X'') is separable if each of its irreducible factors in K[X] have distinct roots in an [[algebraic closure]] of ''K'', that is, if each of its irreducible factors is square-free.<ref>N. Jacobson, Basic Algebra I, p. 233</ref> In this definition, separability depends on the field ''K'', as for example an irreducible polynomial ''P'' which is not separable becomes separable over the splitting field of ''K''. Also, for this definition every polynomial over a [[perfect field]] is separable, which includes in particular all fields of [[characteristic (algebra)|characteristic]] 0, and all [[finite field]]s.  
  <li>[http://bqlchaoshi.com/news/html/?72396.html http://bqlchaoshi.com/news/html/?72396.html]</li>
 
 
Both definitions coincide in case ''P''(''X'') is [[irreducible polynomial|irreducible]] over ''K'', which is the case used to define the notion of a [[separable extension]] of ''K''.
  <li>[http://www.gzhotpower.com/news/html/?17280.html http://www.gzhotpower.com/news/html/?17280.html]</li>
 
 
In the remainder of this article, only the first definition is used.
  <li>[http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/ http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/]</li>
 
 
A polynomial ''P''(''X'') is separable if and only if it is [[coprime]] to its [[formal derivative]] ''P''&prime;(''X'').
  <li>[http://www.dailyqr.com/blog_entry.php?user=1173753&blogentry_id=16103537 http://www.dailyqr.com/blog_entry.php?user=1173753&blogentry_id=16103537]</li>
 
 
==Separable field extensions==
  <li>[http://48.gxhc360.com/forum.php?mod=viewthread&tid=10669 http://48.gxhc360.com/forum.php?mod=viewthread&tid=10669]</li>
Separable polynomials are used to define [[separable extension]]s: A field extension <math>K \subset L</math> is a separable extension if and only if for every <math>\alpha\in L</math>, which is algebraic over ''K'', the [[minimal polynomial (field theory)|minimal polynomial]] of <math>\alpha</math> over ''K'' is a separable polynomial.
 
 
</ul>
[[Inseparable extension]]s (that is extensions which are not separable) may occur only in [[characteristic p|characteristic ''p'']].  
 
The criterion above leads to the quick conclusion that if ''P'' is irreducible and not separable, then ''P''&prime;(''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> &minus; ''T''
with ''K'' the field of [[rational function]]s 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 ''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 that is a product of fields (so, not a 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 coefficients and ''p'' be a prime number which does not divides the leading coefficient of ''P''. Let ''Q'' be the polynomial over the [[finite field]] with ''p'' elements, which is obtained by reducing [[modulo (jargon)|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 [[Cycle (mathematics)|cycles]] of some [[permutation]] of the [[Galois group]] of ''P''.  
 
Another example: ''P'' being as above, a '''resolvent''' ''R'' for a [[Group (mathematics)|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 <math>X^2-D</math> is a resolvent for the [[alternating group]]. This resolvent is always separable if ''P'' is irreducible, but most resolvents are not always separable.
 
==See also==
*[[Frobenius endomorphism]]
 
==References==
{{Reflist}}
* Pages 240-241 of {{Lang Algebra|edition=3}}
 
[[Category:Field theory]]
[[Category:Polynomials]]

Latest revision as of 22:41, 24 November 2014

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