Talk:Normed division algebra

Please note that the article still contains a severe error: R,H,C,O are not the only normed real division algebras. In the article on division algebras, neither associativity, nor existence of an identity, is assumed. The theorem by Hurwitz, as is written in the article referred to, speaks of normed division algebras with unity.

The problem of classifying all normed real division algebras is only partially solved; in one dimension, there is only R; in two dimensions, there are four non-isomorphic algebras. In four dimension, there are infinitely many and in eight, no classification is known.

Please synchronize definitions across Wikipedia articles on the subject. Working with non-associative algebras, I do not assume an algebra to be associative, nor to have an identity. However, I will not change anything, fearing that there will be mismatches between different articles. 130.238.58.123 (talk) 15:52, 11 February 2011 (UTC)


 * At composition algebra where this topic now lives, we've been sure to include "unital." That's in line with our current practice of including identity, and that's how the theorem's been stated everywhere I've seen it. Rschwieb (talk) 18:52, 24 July 2014 (UTC)

The following was the original article.

A division algebra that is also a normed vector space.

The normed division algebras are: These are the only possible normed division algebras.
 * the Real numbers
 * the Complex numbers
 * the Quaternions
 * the Octonions

The entry is clearly false as it stands, but it seems to have some hidden core of truth. How can we rescue the article? --AxelBoldt

(This is a belated response, as I didn't see this article in Recent Changes.) This result is given in John Baez's octonion article. I've never checked it myself, but I believe it's correct if you use the definition of "normed division algebra" that Baez uses: a finite-dimensional unital algebra over R that is also a normed vector space satisfying ||ab|| = ||a||.||b||. This definition appears unreasonably restrictive in disallowing infinite-dimensional algebras and algebras over C. It also conflicts with your definition of division algebra, which requires associativity. I'm not sure what to do about the article. --Zundark, 2002 Jan 9

Would it be better to omit associativity from our definition of division algebra? After all, we already have division ring. --AxelBoldt


 * Yes, I think we should do that. But I'm not sure what the definition should be - there are inequivalent definitions that reduce to our current definition in the associative case. I think "algebra in which the nonzero elements form a loop" is probably what we want, but it would be a good idea to check some books first to see how different authors define it. --Zundark, 2002 Jan 10

I don't claim to understand what you are talking about, but if different authors use these terms with different definitions, we should note that in the article, even if we decide to adopt one of those definitions for Wikipedia's own use. -- SJK


 * Yes, I usually try to do this if there is differing usage. --Zundark, 2002 Jan 10

I'm using a book at the moment, On Quaternions and Octonions, which uses the term composition algebra in the proof of Hurwitz's theorem, which seems to be normally stated for normed division algebras - so I'm editting the definition to have this as an alternative name, and I'll re-direct an article called Composition Algebra to this page. (MatthewMain 20:47, 29 April 2006 (UTC))

I'm sorry, there is now an article called Composition Algebra and has been for 3 months! Serves me right for not checking really... (MatthewMain 20:54, 29 April 2006 (UTC))

Finite/infinite
Can someone confirm (with cite) that there are no infinte NDAs over the reals? Then we can change the William Rowan Hamilton article to reflect that. Rich Farmbrough, 12:41 12 October 2006 (GMT).