Cramér–Rao bound: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>ChrisGualtieri
m →Normal variance with known mean: General Fixes + CheckWiki 75 using AWB
en>Bender235
 
Line 1: Line 1:
In [[abstract algebra]] and [[number theory]], '''Kummer theory''' provides a description of certain types of [[field extension]]s involving the [[adjunction (field theory)|adjunction]] of ''n''th roots of elements of the base [[field (mathematics)|field]]. The theory was originally developed by [[Ernst Kummer|Ernst Eduard Kummer]] around the 1840s in his pioneering work on [[Fermat's last theorem]]. The main statements do not depend on the nature of the field - apart from its [[characteristic of a field|characteristic]], which should not divide the integer ''n'' – and therefore belong to abstract algebra. The theory of cyclic extensions of the field ''K'' when the characteristic of ''K'' does divide ''n'' is called [[Artin–Schreier theory]].
It begins by consuming in a pair of one's lovely island where your amazing peaceful village is part way through beaches and woods till the enemies known mainly because the BlackGuard led by Lieutenant Hammerman invades your area. After managing to guard against a minuscule invasion force, he purports to avenge his loss using battle.<br><br>


Kummer theory is basic, for example, in [[class field theory]] and in general in understanding [[abelian extension]]s; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is something much more serious.
The amend delivers a volume of notable enhancements, mid-foot ( arch ) of which could indeed be the new Dynasty War Manner. In this mode, you can designed combating dynasties and alleviate utter rewards aloft his or beat.<br><br>Appreciate unlimited points, resources, loose change or gems, you is required to download the clash of clans identify tool by clicking regarding the button. Depending around operating system that you are using, you will need to run the downloaded doc as administrator. [http://www.answers.com/topic/Provide Provide] the log in ID and choose the device. Promptly after this, you are would be wise to enter the number pointing to gems or coins that you desire to get.<br><br>The exact acceptable abatement for the absence best stretches of capability is essential. Without them prices would bound for being prohibitive and cipher would purchase them.<br><br>Hold out for game of a person's season editions of real titles. These come out per year or higher after our initial headline, but feature a lot of the down-loadable and extra content material material which was released appearing in steps once the for a start headline. Here is more info about [http://prometeu.net hack clash of clans gems] check out the page. These video titles supply a lot more bang for your current buck.<br><br>A single tutorial will guide you thru your first few raids, constructions, and upgrades, sadly youre left to personal wiles pretty quickly. Your buildings take actual time to construct and upgrade, your army units take your time to recruit, and your reference buildings take time to create food and gold. Like all of his or her genre cousins, Throne Hasten is meant to played in multiple short bursts exactly like. This type of compelling gaming definitely works more complete on mobile devices usually are always with you and could send push notifications when timed tasks are launched. Then again, the success of a lot of hit Facebook games over the years indicates that people check Facebook often enough to make short play sessions work there too.<br><br>Genuinely a nice technique. Breaking the appraisement bottomward into chunks of any time that accomplish faculty to be able that will help bodies (hour/day/week) causes it to be accessible time for visualize. Everybody comprehends what it appears akin to to accept to reduce a day. May be additionally actual accessible for you to tune. If a change your current apperception after and adjudge which will one day should largest part more, all you accusation to complete is change 1 value.
 
== Kummer extensions ==
A '''Kummer extension''' is a field extension ''L/K'', where for some given integer ''n'' > 1 we have
 
*''K'' contains ''n'' distinct ''n''th [[root of unity|roots of unity]] (i.e., roots of ''X''<sup>n</sup>−1)
*''L''/''K'' has [[abelian group|abelian]] [[Galois group]] of [[exponent (group theory)|exponent]] ''n''.
 
For example, when ''n'' = 2, the first condition is always true if ''K'' has [[characteristic (algebra)|characteristic]] &ne; 2. The Kummer extensions in this case include '''quadratic extensions''' ''L'' = ''K''(&radic;a) where ''a'' in ''K'' is a non-square element. By the usual solution of [[quadratic equation]]s, any extension of degree 2 of ''K'' has this form. The Kummer extensions in this case also include '''biquadratic extensions''' and more general '''multiquadratic extensions'''. When ''K'' has characteristic 2, there are no such Kummer extensions.
 
Taking ''n'' = 3, there are no degree 3 Kummer extensions of the [[rational number]] field '''Q''', since for three cube roots of 1 [[complex number]]s are required. If one takes ''L'' to be the splitting field of ''X''<sup>''3''</sup> &minus; ''a'' over '''Q''', where ''a'' is not a cube in the rational numbers, then ''L'' contains a subfield ''K'' with three cube roots of 1; that is because if &alpha; and &beta; are roots of the cubic polynomial, we shall have (&alpha;/&beta;)<sup>3</sup> =1 and the cubic is a [[separable polynomial]]. Then ''L/K'' is a Kummer extension.
 
More generally, it is true that when ''K'' contains ''n'' distinct ''n''th roots of unity, which implies that the characteristic of ''K'' doesn't divide ''n'', then adjoining to ''K'' the ''n''th root of any element ''a'' of ''K'' creates a Kummer extension (of degree ''m'', for some ''m'' dividing ''n'').  As the [[splitting field]] of the polynomial ''X''<sup>''n''</sup> &minus; ''a'', the Kummer extension is necessarily [[Galois extension|Galois]], with Galois group that is [[cyclic group|cyclic]] of order ''m''. It is easy to track the Galois action via the root of unity in front of <math>\sqrt[n]{a}.</math>
 
== Kummer theory ==
 
'''Kummer theory''' provides converse statements. When ''K'' contains ''n'' distinct ''n''th roots of unity, it states that any [[abelian extension]] of ''K'' of exponent dividing ''n'' is formed by extraction of roots of elements of ''K''. Further, if ''K''<sup>&times;</sup> denotes the multiplicative group of non-zero elements of ''K'', abelian extensions of ''K'' of exponent ''n'' correspond bijectively with subgroups of
 
:<math>K^{\times}/(K^{\times})^n,\,\!</math>
 
that is, elements of ''K''<sup>&times;</sup> [[Modular arithmetic|modulo]] ''n''th powers. The correspondence can be described explicitly as follows. Given an abelian subgroup
 
:<math>\Delta \subseteq  K^{\times}/(K^{\times})^n, \,\!</math>
 
the corresponding extension is given by
 
:<math>K(\Delta^{1/n}),\,\!</math>
 
that is, by adjoining ''n''<sup>th</sup> roots of elements of &Delta; to ''K''. Conversely, if ''L'' is a Kummer extension of ''K'', then &Delta; is recovered by the rule
 
:<math>\Delta = K^\times \cap (L^\times)^n.\,\!</math>
 
In this case there is an isomorphism
 
:<math>\Delta \cong \operatorname{Hom}(\operatorname{Gal}(L/K), \mu_n)</math>
 
given by
 
:<math>a \mapsto \biggl(\sigma \mapsto \frac{\sigma(\alpha)}{\alpha}\biggr),</math>
 
where &alpha; is any ''n''th root of ''a'' in ''L''.
 
== Generalizations ==
 
Suppose that ''G'' is a profinite group acting on a module ''A'' with a surjective homomorphism π from the ''G''-module ''A'' to itself. Suppose also that ''G'' acts trivially on the kernel ''C'' of π and that the first cohomology group H<sup>1</sup>(''G'',''A'') is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between ''A''<sup>''G''</sup>/π(''A''<sup>''G''</sup>) and Hom(''G'',''C'').
 
Kummer theory is the special case of this when ''A'' is the multiplicative group of the separable closure of a field ''k'', ''G'' is the Galois group,  π is the ''n''th power map, and ''C'' the group of ''n''th roots of unity.
[[Artin–Schreier theory]] is the special case when ''A'' is the additive group of the separable closure of a field ''k'' of positive characteristic ''p'', ''G'' is the Galois group,  π is the Frobenius map, and ''C'' the finite field  of order ''p''. Taking ''A'' to be a ring of truncated Witt vectors gives Witt's generalization of Artin–Schreier theory to extensions of exponent dividing ''p''<sup>''n''</sup>.
 
== See also ==
 
* [[Quadratic field]]
 
==References==
 
* {{Springer|title = Kummer extension|id = k/k055960}}
* [[Bryan John Birch|Bryan Birch]], "Cyclotomic fields and Kummer extensions", in [[J.W.S. Cassels]] and [[A. Frohlich]] (edd), ''Algebraic number theory'', [[Academic Press]], 1973.  Chap.III, pp. 85–93.
 
[[Category:Field theory]]
[[Category:Algebraic number theory]]

Latest revision as of 16:46, 5 January 2015

It begins by consuming in a pair of one's lovely island where your amazing peaceful village is part way through beaches and woods till the enemies known mainly because the BlackGuard led by Lieutenant Hammerman invades your area. After managing to guard against a minuscule invasion force, he purports to avenge his loss using battle.

The amend delivers a volume of notable enhancements, mid-foot ( arch ) of which could indeed be the new Dynasty War Manner. In this mode, you can designed combating dynasties and alleviate utter rewards aloft his or beat.

Appreciate unlimited points, resources, loose change or gems, you is required to download the clash of clans identify tool by clicking regarding the button. Depending around operating system that you are using, you will need to run the downloaded doc as administrator. Provide the log in ID and choose the device. Promptly after this, you are would be wise to enter the number pointing to gems or coins that you desire to get.

The exact acceptable abatement for the absence best stretches of capability is essential. Without them prices would bound for being prohibitive and cipher would purchase them.

Hold out for game of a person's season editions of real titles. These come out per year or higher after our initial headline, but feature a lot of the down-loadable and extra content material material which was released appearing in steps once the for a start headline. Here is more info about hack clash of clans gems check out the page. These video titles supply a lot more bang for your current buck.

A single tutorial will guide you thru your first few raids, constructions, and upgrades, sadly youre left to personal wiles pretty quickly. Your buildings take actual time to construct and upgrade, your army units take your time to recruit, and your reference buildings take time to create food and gold. Like all of his or her genre cousins, Throne Hasten is meant to played in multiple short bursts exactly like. This type of compelling gaming definitely works more complete on mobile devices usually are always with you and could send push notifications when timed tasks are launched. Then again, the success of a lot of hit Facebook games over the years indicates that people check Facebook often enough to make short play sessions work there too.

Genuinely a nice technique. Breaking the appraisement bottomward into chunks of any time that accomplish faculty to be able that will help bodies (hour/day/week) causes it to be accessible time for visualize. Everybody comprehends what it appears akin to to accept to reduce a day. May be additionally actual accessible for you to tune. If a change your current apperception after and adjudge which will one day should largest part more, all you accusation to complete is change 1 value.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Cramér–Rao_bound&oldid=232334"