From: mann@vms.huji.ac.il (Avinoam Mann)
Newsgroups: sci.math.research
Subject: Re: Generalized Theorem of Frobenius
Date: 26 Aug 1998 03:10 IDT
News-Software: VAX/VMS VNEWS 1.41    

In article <6rrg2q$d3c$1@nnrp1.dejanews.com>, aj_dude@my-dejanews.com writes...
>The classical Theorem of Frobenius says that there are exactly three (up to
>an isomorphism) associative division R-algebras (where R denotes the field of
>real numbers): - R  itself, - C  (complex numbers), - H (Hamilton's
>quaternions).
> 
>There is a well-known generalization of this statement onto non-associative
>case: namely, if one substitutes "associative" by "alternative" above, there
>are four such algebras: R,  C,	H and - O (Caley's octaves).
> 
>(Recall that an algebra is _alternative_ iff for each its two elements $u$
>and $v$ one has  u(vv) = (uv)v	and  (uu)v = u(uv). An algebra is alternative
>iff its each subalgebra generated by two elements, is associative -- that is
>Artin's lemma).
> 
>My question is the following. Does anybody knows who first proved the second
>statement. I need this for preparation a course for students.

Hurwitz in 1898 (Nach. Ges. d. Wiss. Goettingen, 309-316) (according to
Eilenberg-Steenrod, Foundations of Algebraic Topology, p. 320).

                                                           Avinoam Mann

 
>I know the further generalizations of the theorem (concerning Caley-Dixon
>process), but particulary interested to know the name of the author of this
>result concerning classical  R-algebras  R, C, H and O.
> 
>Thank you.
>--
> AJ, Dr. math.
>posting dude
> 
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==============================================================================

From: wmoehri@gwdg.de (Willi Moehring)
Newsgroups: sci.math.research
Subject: Re: Generalized Theorem of Frobenius
Date: 26 Aug 1998 10:21:06 +0200

mann@vms.huji.ac.il (Avinoam Mann) writes:
[quote of previous article deleted -- djr]

According to the book of Eddinghaus, Hermes, Hirzebruch, ...  "Zahlen"
the theorm of Hurwitz refers to "composition algebras". The theorem about
associative algebras is in that book attributed to Zorn (1933). There, however
the additional condition of being "quadratic" is required.

                     Willi Mohring
==============================================================================

From: mann@vms.huji.ac.il (Avinoam Mann)
Newsgroups: sci.math.research
Subject: Re: Generalized Theorem of Frobenius
Date: 29 Aug 1998 05:19 IDT

In article <6s0ghi$arp$1@gwdu19.gwdg.de>, wmoehri@gwdg.de (Willi Moehring) 
writes...
[quote of previous article deleted -- djr]

An algebra over R is "normed" if it satisfies N(xy) = N(x)N(y), where N(x)
is the Euclidean norm of x. It's a surprising fact that a dvision algebra
over R is alternative iff it is normed. A composition algebra is more general
than a normed one, in that that the norm equation above is replaced by
P(xy) = Q(x)R(y) for some quadratic forms P,Q,R; I don't recall right now if
they are supposed also to be positive definite. I believe that it's true that
Hurwitz proved his result for normed algebras; I'm not sure who first noted
the equivalence with alternative algebras; maybe it was Zorn, his name is
linked to non-associative algebra. I don't know which result of Zorn the book
by Ebbinghaus et al refers to; from the formulation given by Willi Mohring it
sems possible that it refers to division algebras in which each element has
a quadratic (or linear) minimal polynomial.
I'm going away to-morrow, so I'll not be able to check the literature before
about ten days time.
                                                           Avinoam Mann