September 11 attacks and Characteristic (algebra): Difference between pages

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In [[abstract algebra]], the '''characteristic''' of a [[mathematical ring|ring]] ''R'' is defined to be the smallest
<table width="100%"><tr bgcolor="#cccccc" align="center"><td>
positive [[integer]] ''n'' such that 1<sub>''R''</sub>+...+1<sub>''R''</sub>
<b> [[September_11,_2001_Terrorist_Attack/In Memoriam|In Memoriam, September 11, 2001]]</b>
(with ''n'' summands) yields 0. If no such ''n'' exists, we say that the
</td></tr></table>
characteristic of ''R'' is 0.
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Alternatively, the characteristic of the ring ''R'' may be defined as that unique [[natural number]] ''n'' such that ''R'' contains a [[subring]] [[ring homomorphism|isomorphic]] to the factor ring '''Z'''/''n'''''Z'''.
On the morning of [[September 11]], [[2001]], what might well be the most devastating [[terrorist incidents|terrorist attack]] in the history of the world occurred concurrently in [[New York City]], [[Washington, D.C.]] and near [[Pittsburgh]]. Four passenger jets were [[September_11,_2001_Terrorist_Attack/Hijackers|hijacked]] and then deliberately crashed into the [[World Trade Center]] and the [[Pentagon]]. Both towers of the World Trade Center subsequently collapsed, and part of the Pentagon was destroyed in the ensuing fire. [[September_11,_2001_Terrorist_Attack/Casualties|Casualties]] are expected to be in the thousands: 265 on the planes; about 3000 people (early estimates ranged as high as 6500 people), including hundreds of firefighters who had rushed in, at the World Trade Center; and 125 at the Pentagon. The third hijacked plane was taken down in a Pennsyvania field by the heroic efforts of passengers and crew.


'''Examples and notes:'''
Some passengers on the doomed flights were able to make phone calls reporting on events on board. They reported that there were more than one hijacker on each plane and that they took control of the planes using [[box-cutter knife|box-cutter knives]]. It appears that the passengers on the fourth jet tried to overpower the hijackers and that the plane crashed in a sparsely populated area as a result, thereby missing its intended target, which may have been the [[White House]].


* If ''R'' and ''S'' are rings and there exists a [[ring homomorphism]] ''R'' <tt>-></tt> ''S'', then the characteristic of ''S'' divides the characteristic of ''R''.
The attack had immediate and deep global [[September_11,_2001_Terrorist_Attack/World political effects|political effects]] and [[September_11,_2001_Terrorist_Attack/World economic effects|economic effects]] and inspired an international outpouring of [[September_11,_2001_Terrorist_Attack/Memorials and Services|memorials and services]]. The massive undertaking of [[September_11,_2001_Terrorist_Attack/Rescue and recovery effort|rescue and recovery]], and of providing [[September_11,_2001_Terrorist_Attack/Assistance|assistance]] to the [[September_11,_2001_Terrorist_Attack/Survivors|survivors]] and victims, is ongoing. There will be a great need for [[September_11,_2001_Terrorist_Attack/Donations|donations]] for a long time.
* For any [[integral domain]] (and in particular for any [[field]]), the characteristic is either 0 or [[prime number|prime]].
* For any [[ordered field]] (for example, the [[rational number|rationals]] or the [[real number|reals]]) the characteristic is 0.
* The ring '''Z'''/''n'''''Z''' of integers [[modular arithmetic|modulo]] ''n'' has characteristic ''n''.
* If ''R'' is a [[subring]] of ''S'', then ''R'' and ''S'' have the same characteristic. For instance, if ''q''(''X'') is a prime [[polynomial]] with coefficients in the field '''Z'''/''p'''''Z''' where ''p'' is prime, then the factor ring ('''Z'''/''p'''''Z''')[''X'']/(''q''(''X'')) is a field of characteristic ''p''. Since the [[complex number|complex numbers]] contain the rationals, their characteristic is 0.
* Any field of 0 characteristic is infinite. The [[finite field]] GF(''p''<sup>''n''</sup>) has characteristic ''p''.
* There exist infinite fields of prime characteristic. For example, the field of all rational functions over '''Z'''/''p'''''Z''' is one such. The [[algebraic closure]] of '''Z'''/''p'''''Z''' is another example.
* The size of any finite field of characteristic ''p'' is a power of ''p''. Since in that case it must contain '''Z'''/''p'''''Z''' it must also be a [[vector space]] over that field and from [[linear algebra]] we know that the sizes of finite vector spaces over finite fields are a power of the size of the field.
* This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size ''p''<sup>''n''</sup>. So its size is (''p''<sup>''n''</sup>)<sup>''m''</sup> = ''p''<sup>''nm''</sup>. QED)
* If an integral domain ''R'' has prime characteristic ''p'', then we have (''x'' + ''y'')<sup>''p''</sup> = ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> for all elements ''x'' and ''y'' in ''R''. The map ''f''(''x'') = ''x''<sup>''p''</sup> defines a [[injective, surjective and bijective functions|injective]] [[ring homomorphism]] ''R'' <tt>-></tt> ''R''. It is called the ''Frobenius homomorphism''.
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<b>Characteristic</b> is also sometimes used as a piece of jargon in discussions of [[Universal--metaphysics|universals]] in [[metaphysics]], often in the phrase 'distinguishing characteristics'.


Though no group has claimed [[September_11,_2001_Terrorist_Attack/Responsibility|responsibility]], the U.S. government immediately launched a full-scale [[September_11,_2001_Terrorist_Attack/US governmental response|response]], stating its intentions to go to war against those responsible. On October 7, a coalition led by the United States launched an attack in Afghanistan. See '''[[2001 U.S. Attack on Afghanistan]]'''.

Following the attack, the United States has been on heightened alert for new terrorist attacks. In late September, [[cases of anthrax]] started breaking out; although, no connection to the September 11 attack has been found. See '''[[2001 anthrax attack]]'''.

=== Timeline: [[September_11,_2001_Terrorist_Attack/Timeline|September]] - [[September_11,_2001_Terrorist_Attack/Timeline October 2001|October]] - [[September_11,_2001_Terrorist_Attack/Timeline November 2001|November]] - [[September_11,_2001_Terrorist_Attack/Timeline December 2001|December]] - [[September_11,_2001_Terrorist_Attack/Back history|Back history]] ===

A minute-by-minute breakdown of events, and a history of events leading up to the attack. Where to [http://www.wikipedia.com/wiki.cgi?action=edit&id=September_11,_2001_Terrorist_Attack/Timeline_October_2001 add] or [[September_11,_2001_Terrorist_Attack/Timeline_October_2001|read]] the latest breaking news.

=== [[September_11,_2001_Terrorist_Attack/Casualties|Casualties]] - [[September_11,_2001_Terrorist_Attack/Missing Persons|Missing Persons]] - [[September_11,_2001_Terrorist_Attack/Survivors|Survivors]] - [[September_11,_2001_Terrorist_Attack/Personal experiences|Personal experiences]] ===

Victims and survivors, and their personal stories. All together, these pages may serve as a memorial to those lost. Your help is welcome and needed.

=== [[September_11,_2001_Terrorist_Attack/Give Blood|Give Blood]] - [[September_11,_2001_Terrorist_Attack/Donations|Donations]] - [[September_11,_2001_Terrorist_Attack/Assistance|Assistance]] - [[September_11,_2001_Terrorist_Attack/Rescue and recovery effort|Rescue and Recovery Effort]] ===

How people across the world have helped, and how they can continue to do so. What is being done to assist victims.

=== [[September_11,_2001_Terrorist_Attack/Closings and Cancellations|Closings and Cancellations]] - [[September_11,_2001_Terrorist_Attack/Memorials and Services|Memorials and Services]] - [[September_11,_2001_Terrorist_Attack/US governmental response|US Governmental Response]] ===

How people have responded.

=== [[September_11,_2001_Terrorist_Attack/Responsibility|Responsibility]] - [[September_11,_2001_Terrorist_Attack/Hijackers|Hijackers]] - [[September_11,_2001_Terrorist_Attack/World political effects|World political effects]] - [[September_11,_2001_Terrorist_Attack/World economic effects|World economic effects]] ===

The long-term history of the attack.

=== [[September_11,_2001_Terrorist_Attack/Airport security|Airport security]] - [[September_11,_2001_Terrorist_Attack/Opportunists|Opportunists]] - [[September_11,_2001_Terrorist_Attack/Communications|Communications]] - [[September_11,_2001_Terrorist_Attack/Slogans and terms|Slogans and terms]] ===

Specific topics.


See also: [[World Trade Center]] -- [[Pentagon]] -- [[New York City]] -- [[Washington, D.C.]] -- [[American Airlines flight 11|AA Flight 11]] -- [[United Airlines flight 175|UA Flight 75]] -- [[American Airlines flight 77|AA Flight 77]] -- [[United Airlines flight 93|UA Flight 93]] -- [[United States/Department of Defense|U.S. Department of Defense]] -- [[FBI]] -- [[terrorism]] -- [[domestic terrorism]] -- [[terrorist incidents]] -- [[Osama bin Laden]] -- [[Taliban]] -- [[Islamism]] -- [[Afghanistan]] -- [[collective trauma]] -- [[September 11]]
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[[talk:Characteristic|/Talk]]


=== [[September_11,_2001_Terrorist_Attack/External news sites|External news sites and references]] - [[September_11,_2001_Terrorist_Attack/Misinformation and rumors|Misinformation and Rumors]] ===

A view on why Arabs hate the US: http://newuntitledpage.saburovo.com/rants.php?to_view=55

[[talk:September_11,_2001_Terrorist_Attack|Talk]] - [[September_11,_2001_Terrorist_Attack/Casualties_Talk|Casualties_Talk]] - [[talk:Give Blood]] - [[September_11,_2001_Terrorist_Attack/Hijackers_Talk|Hijackers_Talk]]

[[September_11,_2001_Terrorist_Attack/Casualties template|Casualties template]] - [[September_11,_2001_Terrorist_Attack/Footer template|Footer template]]

Revision as of 08:55, 3 February 2002

In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1R+...+1R (with n summands) yields 0. If no such n exists, we say that the characteristic of R is 0.

Alternatively, the characteristic of the ring R may be defined as that unique natural number n such that R contains a subring isomorphic to the factor ring Z/nZ.

Examples and notes:

  • If R and S are rings and there exists a ring homomorphism R -> S, then the characteristic of S divides the characteristic of R.
  • For any integral domain (and in particular for any field), the characteristic is either 0 or prime.
  • For any ordered field (for example, the rationals or the reals) the characteristic is 0.
  • The ring Z/nZ of integers modulo n has characteristic n.
  • If R is a subring of S, then R and S have the same characteristic. For instance, if q(X) is a prime polynomial with coefficients in the field Z/pZ where p is prime, then the factor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, their characteristic is 0.
  • Any field of 0 characteristic is infinite. The finite field GF(pn) has characteristic p.
  • There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ is one such. The algebraic closure of Z/pZ is another example.
  • The size of any finite field of characteristic p is a power of p. Since in that case it must contain Z/pZ it must also be a vector space over that field and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field.
  • This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pn. So its size is (pn)m = pnm. QED)
  • If an integral domain R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R. The map f(x) = xp defines a injective ring homomorphism R -> R. It is called the Frobenius homomorphism.

Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.


/Talk