Re: on quaternions and octonions

From: John Baez (baez_at_galaxy.ucr.edu)
Date: 08/06/04 

Date: Fri, 6 Aug 2004 11:29:41 +0000 (UTC)

I'm sorry it's taken me so long to get back to this thread.
I should thank Bill Dubuque for his contribution! I was in
Dublin, where Hamilton invented the quaternions and Hawking
conceded his famous bet on black hole information. For some
pictures, see:

http://math.ucr.edu/home/baez/dublin/

Anyway....

In article <y8z658qhumd.fsf@nestle.csail.mit.edu>,
Bill Dubuque <wgd@nestle.csail.mit.edu> wrote:

> John Baez wrote:

>> Jose Carlos Santos wrote:

>>> JB had written:
 
>>>> 3) What is the precise statement of their unique prime
>>>> factorization theorem for Hurwitz integral quaternions?

>I suppose that you mean this theorem: "To any factorization of the norm
>q of a primitive Hurwitz integer Q into a product p_0p_1 ... p_k of
>rational primes, there is a factorization
>
> Q = P_0P_1 ... P_k
>
>of Q into a product of Hurwitz primes with N(P_0) = p_0, ..., N(P_k) =
>p_k. We shall say that 'the factorization P_0P_1 ... P_k is modelled on
>the facorization p_0p_1 ... p_k of N(Q).' Moreover, if Q =
>P_0P_1 ... P_k is any one factorization modelled on p_0p_1 ... p_k,
>then the others have the form
>
> Q = P_0U_1.U_1^{-1}P_1U_2. ... .U_k^{-1}P_k
>
>i.e., 'the factorization on a given model is unique up to
>unit-migration.'"

By the way, I'm pretty sure the "norm" of the quaternion

a + bi + cj + dk

is defined to be

a^2 + b^2 + c^2 + d^2

but if not, someone had better tell me before I make a fool
of myself in print.

Now, about this:

>> I'd be very happy if anyone could show that "factorization
>> on a given model is not unique up to unit-migration" for the
>> Lipschitz integral quaternions, where the quoted phrase is
>> defined as above, and the Lipschitz integral quaternions are
>> simply those of the form
>>
>> a + b + cj + dk
>>
>> where a,b,c,d are integers.
>>
>> I'll cite the 5 first people who come up with counterexamples!

>Lipschitz pointed out examples himself according to Fenster's
>interesting historical article [1].

Thanks for the reference! I'll cite you, and Fenster, and Lipschitz...

> "[..] To factor, A = -1+3i+1j+2k, for example, he found triples
> satisfying x^2 + y^2 + z^2 = 0 (mod 3) and x^2 + y^2 + z^2 = 0 (mod 5),
> determined the associated integral quaternions with norms three and five,
> and obtained the two factorizations
>
> -1 + 3i + 1j + 2k = (1 + 2i)(1 + i + j) = -(1 - i + j)(1 - 2k).
>
> The above equation not only illustrates LIPSCHITZ's comment regarding
> the importance of the order of the factors but also shows the absence
> of unique factorization in his proposed system of integral
> quaternions.

... even though this is *not* an example of failure of "unique
factorization up to unit-migration" in Conway and Smith's sense,
because the two factorizations above are "modeled after" two different
factorizations of the number (-1)^2 + 3^2 + 1^2 + 2^2 = 15, namely
as (1^2 + 2^2)(1^2 + 1^1 + 1^2) = 5 x 3, and as 3 x 5.

Here's an example of the sort of nonuniqueness I was looking for:

2 x 2 = (1 + i + j + k)(1 - i - j - k)

This is related to this remark:

> The lack of unique factorization seems to grow
> worse when four divides the norm of the integral quaternion in
> question. In this case, with twenty-four renditions of an integral
> prime quaternion of norm 4, there are equally many factorizations.

(which only makes sense if "norm" is defined as I mentioned above).

It "grows worse" in this way precisely because we need the
Hurwitz integral quaternions get uniqueness up to unit-migration,
and we have phenomena like

2U = (1 + i + j + k)
U^{-1}2 = (1 - i - j - k)

where U is a unit in the Hurwitz integral quaternions (of which
there are 24).

>[1] Fenster, Della Dumbaugh (1-RICH-CM)
>Leonard Eugene Dickson and his work in the arithmetics of algebras.
>Arch. Hist. Exact Sci. 52 (1998), no. 2, 119-159.

Thanks for the reference! I should read it!

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