29 (number): Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Arthur Rubin
Reverted good faith edits by Oatmealcookiemon: Untracable statement; is "Twentynine" a group or an album, and the specific _song_ on the album shouldn't be there. Removing entirely. (TW)
en>ClueBot NG
m Reverting possible vandalism by 50.203.217.98 to version by Richard Yin. False positive? Report it. Thanks, ClueBot NG. (2039883) (Bot)
 
Line 1: Line 1:
In [[mathematics]], two slightly different notions of '''separable polynomial''' are used, by different authors.
== Oakley Sunglasses Nz Buy I think a nice ==


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]].
I think a nice, warm headband that you can kind of throw around anything. And, a good street bike jacket. The thing that I like with regards to article marketing is that you don't have to write any [http://www.lordens.co.nz/webalizer/eventwin.asp?o=44-Oakley-Sunglasses-Nz-Buy Oakley Sunglasses Nz Buy] additional content to publish in directories if you don't want to. For me personally I write often in my [http://www.milfordflights.co.nz/includes/flights.asp?r=53-Cheap-Nike-Roshe-Run-Mens Cheap Nike Roshe Run Mens] blog, so [http://www.thebluepub.co.nz/templates/Express/green/enquiries.asp?k=3-Nike-Shop-Auckland Nike Shop Auckland] all I do will be wait for my posts to often be indexed by Google and then add them to the top 5 directories which has a link back to my lead capture page and website.. <br><br>The one who is definitely running the state is Scott Madigan. He's the one who told them go for it .. According to recent reports, the 's Qaeda terror organization has been decimated and nearly all its command terminated. Remaining remnants of the group are now believed to have a home in the country of Yemen, frantically circling about that country on camel back hopes of avoiding being spotted and blown up by predator drones. <br><br>Most of these select guests got very first crack at the wares as well as the luxury of browsing unencumbered by way of the crowd at the MCA party and the $12 a ticket lookers who (it is hoped) will swarm the location this weekend. The MCA's "patron reception,In . still open to anyone happy to cough up $400 $600 a ticket, did not begin till 4 Pm hours, two hours after the first critique closed. <br><br>I remember a few years ago I somehow managed to code a simple MP3 and PDF obtain service from scratch, using Pay pal to process transactions. It was a fair amount of work and I had a good time doing it. Do not let him lay claim one of your daughters, he is solely taking money out of your jean pocket. Even if he were to assert them and then give you dollars, some of that money you would possess gotten back anyway in based mostly credits. <br><br>Our limos are usually clean and nice without any with the expensive bells and whistles of different Denver limo companies. Colorado Platinum offers great limos and shuttle packages to provide you to and from all the ski location. What to See Angkor's temple design was heavily influenced by Indian native ideas. From the earliest occasions of the city, Angkor had been conceived as a symbolic universe structured reported by Hindu cosmology. <br><br>Although we do not bar members from attending school part time throughout their service year, it is not proposed. We have an intense program which will take a lot time and energy. If you're over a Mac, go to the bottom proper of your window. There should be the square with 3 straight lines in it under the browse bar. <br><br>The book is separated into regional sections [http://www.thebluepub.co.nz/templates/Express/green/enquiries.asp?k=36-Nike-Shox-Nz-Shoes Nike Shox Nz Shoes]   North east and Mid Atlantic, To the south, Midwest and West South. Each place gets a 2 page bit with images describing the restaurant, its record, owners and their specialty in order to fame. Be aware that the agency will protect them and not give out contact info, probably. And since they're mainly the top stylists people very may well have regular assistants.<ul>
 
 
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 21:41, 24 November 2014

Oakley Sunglasses Nz Buy I think a nice

I think a nice, warm headband that you can kind of throw around anything. And, a good street bike jacket. The thing that I like with regards to article marketing is that you don't have to write any Oakley Sunglasses Nz Buy additional content to publish in directories if you don't want to. For me personally I write often in my Cheap Nike Roshe Run Mens blog, so Nike Shop Auckland all I do will be wait for my posts to often be indexed by Google and then add them to the top 5 directories which has a link back to my lead capture page and website..

The one who is definitely running the state is Scott Madigan. He's the one who told them go for it .. According to recent reports, the 's Qaeda terror organization has been decimated and nearly all its command terminated. Remaining remnants of the group are now believed to have a home in the country of Yemen, frantically circling about that country on camel back hopes of avoiding being spotted and blown up by predator drones.

Most of these select guests got very first crack at the wares as well as the luxury of browsing unencumbered by way of the crowd at the MCA party and the $12 a ticket lookers who (it is hoped) will swarm the location this weekend. The MCA's "patron reception,In . still open to anyone happy to cough up $400 $600 a ticket, did not begin till 4 Pm hours, two hours after the first critique closed.

I remember a few years ago I somehow managed to code a simple MP3 and PDF obtain service from scratch, using Pay pal to process transactions. It was a fair amount of work and I had a good time doing it. Do not let him lay claim one of your daughters, he is solely taking money out of your jean pocket. Even if he were to assert them and then give you dollars, some of that money you would possess gotten back anyway in based mostly credits.

Our limos are usually clean and nice without any with the expensive bells and whistles of different Denver limo companies. Colorado Platinum offers great limos and shuttle packages to provide you to and from all the ski location. What to See Angkor's temple design was heavily influenced by Indian native ideas. From the earliest occasions of the city, Angkor had been conceived as a symbolic universe structured reported by Hindu cosmology.

Although we do not bar members from attending school part time throughout their service year, it is not proposed. We have an intense program which will take a lot time and energy. If you're over a Mac, go to the bottom proper of your window. There should be the square with 3 straight lines in it under the browse bar.

The book is separated into regional sections Nike Shox Nz Shoes North east and Mid Atlantic, To the south, Midwest and West South. Each place gets a 2 page bit with images describing the restaurant, its record, owners and their specialty in order to fame. Be aware that the agency will protect them and not give out contact info, probably. And since they're mainly the top stylists people very may well have regular assistants.

Retrieved from "https://en.formulasearchengine.com/index.php?title=29_(number)&oldid=229115"