Talk:Division algebra

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Mathematics rating: Start Class Mid Priority  Field: Algebra

"In this page, however, we will assume associativity".

huh? Fields and associative algebras are both associative (in multiplication, which is what we're interested in here, right?)

What's the purpose of this statement? Associativity was never at issue.

-- user:clausen

The statement is at the end of a paragraph which gives a different (non-associative) definition. If the statement were omitted, it would be unclear which definition is being used in the rest of the article. --Zundark 08:36 5 Jun 2003 (UTC)

Why do we have separate pages for division ring and division algebra? There isn't much difference.

Waltpohl 07:46, 24 Feb 2004 (UTC)

It seems that 'algebra' here is not always assumed associative.

Charles Matthews 09:12, 24 Feb 2004 (UTC)

Examples would be nice!

rs2 2004.03.10, 01:02 (UTC)

[edit] Field or Ring

Shouldn't the article refer to rings ? The way I read it, all fields are divisional algebras. -- Nic Roets 13:52, 20 July 2005 (UTC)

Yes, all fields are division algebras. From what I know, division algebras are generalizations of fields. Basically, a division algebra is an object in which you can still do division, but it does not need to be commutative or associative. Oleg Alexandrov 15:30, 20 July 2005 (UTC)


Hi

I have edited Example of a non-associative algebra but I must have slipped up somewhere and have seemingly proved that it isn't an algebra at all. I got (ax)*y=\overline{(ax)y}=\overline{a(xy)}=\overline{a}\cdot\overline{xy}=\overline{a}(x*y).

Can anyone patch up my slip in reasoning? Robinh 19:25, 7 August 2005 (UTC)

There is no slip in your reasoning. \overline{a}=a since a is real. In fact this division algebra you give is isomorphic to the usual multiplication on the complex plane, the isomorphism given by complex conjugation. perkinsrc008 12:19, 14 April 2008 (UTC)

[edit] References

How come there are no references here for any proofs? Can anybody point in in the direction of why this is true:

* It is known about the dimension of a finite-dimensional division algebra A over a field K:
   * dim A= 1 if K is algebraically closed,
   * dim A= 1, 2, 4 or 8 if K is real closed  —Preceding unsigned comment added by Moxmalin (talk • contribs) 17:10, 20 March 2008 (UTC)