|
|
| Line 1: |
Line 1: |
| The '''Latimer–MacDuffee theorem''' is a [[theorem]] in [[abstract algebra]], a branch of [[mathematics]].
| | 45 year old Engineering Technologist Houston Earlywine from Holland Landing, spends time with hobbies which includes bell ringing, [http://comoconseguirdinheiro.comoganhardinheiro101.com como ganhar dinheiro na internet] and keep. Gains encouragement by planing a trip to Río Abiseo National Park. |
|
| |
| Let <math>f</math> be a [[monic polynomial|monic]], [[irreducible polynomial]] of degree <math>n</math>. The Latimer–MacDuffee theorem gives a one-to-one correspondence between <math>\mathbb{Z}</math>-[[matrix similarity|similarity classes]] of <math>n\times n</math> [[matrix (mathematics)|matrices]] with [[characteristic polynomial]] <math>f</math> and the [[ideal class]]es in the [[Order (ring theory)|order]]
| |
| | |
| :<math>\mathbb{Z}[x]/(f(x)). \, </math> | |
| | |
| where ideals are considered equivalent if they are equal up to an overall (nonzero) rational scalar multiple. (Note that this order need not be the full ring of integers, so nonzero ideals need not be invertible.) Since an order in a number field has only finitely many ideal classes (even if it is not the maximal order, and we mean here ideals classes for all nonzero ideals, not just the invertible ones), it follows that there are only finitely many [[conjugacy class]]es of matrices over the integers with characteristic polynomial <math>f(x)</math>.
| |
| | |
| ==References==
| |
| * [[Claiborne Latimer|Claiborne G. Latimer]] and [[Cyrus Colton MacDuffee|C. C. MacDuffee]], "A Correspondence Between Classes of Ideals and Classes of Matrices", ''[[Annals of Mathematics]]'', 1933.
| |
| | |
| {{DEFAULTSORT:Latimer-MacDuffee theorem}}
| |
| [[Category:Theorems in abstract algebra]]
| |
| | |
| | |
| {{algebra-stub}}
| |
Revision as of 23:24, 2 March 2014
45 year old Engineering Technologist Houston Earlywine from Holland Landing, spends time with hobbies which includes bell ringing, como ganhar dinheiro na internet and keep. Gains encouragement by planing a trip to Río Abiseo National Park.