Can every polynomial with quaternion coefficients be factored into linear factors?

Factoring quaternion polynomials
--------------------------------

There had been some postings on the net recently about the
factorization of polynomials with coefficients in the quaternions.
Somebody from Princeton (I wish I'd saved these postings) had given a
topological proof of the fact that such polynomials can always be
factored into linear polynomials.  Anyway, I got to thinking that there
ought to be a purely algebraic proof, and while hunting some
references, came across this article by Ivan Niven in the American Math
Monthly vol 48 (1941) pg 654-661, where he proves the remarkable fact
if D is a finite-dimensional non-commutative division algebra over its
center C, then polynomials with coeffs in D factor into linear factors
in D if and only if C is real-closed and D is the algebra of
quaternions over C!

The proof is not his - in one direction (D is the quaternions over
real-closed => polynomials factor) is by Jacobson, and the proof of the
converse is by Baer.  Both proofs are purely algebraic, and the proof
of the converse is especially beautiful (heady Artin-Schreier stuff).

If we apply this theorem to quaternions over the reals, then modulo the
fundamental theorem of algebra (which tells you that R is real-closed),
one gets an algebraic proof of the factorization.

Here is a quick sketch of the algebraic proof that polys in H - the
quaternions over R - factor into linear factors in H.

Let f = sum ( a_i x**i), and consider the polynomial
f~ = sum(a_i~ x**i), where a_i~ is the conjugate of a_i.  (The
conjugate of a+ib+jc+kd is a-ib-jc-kd [Editor: "a-ib-ic-kd" in the
original].)  One checks easily that f*f~ (the product of f and f~) has
its coefficients in R.  This factors into linear factors in the
subfield C of H.  This implies that f then factors in H into linear
factors, the factors being similar to those that appear in the
factorization of f*f~.  Niven quotes some results of O.Ore to prove
this last statement, but it can also be proved as follows:  Let x-c be
the right-most factor of f*f~.  It is well known that if x-c divides a
product g*h of polynomial from the right (i.e., g*h = p*(x-c) for some
polynomial p(x)), and if x-c does not divide h from the right, then
x-c' divides g from the right for some conjugate c' of c.  (This is
true for polynomials over any division ring, and can be proved
easily.)  Now, if x-c divides f~ from the right, then because the map
f|--> f~ is an anti-automorphism of H[x], x-c~ will divide f from the
left, i.e., there is a polynomial p such that f = (x-c~)*p, with
deg p < deg f.  If on the other hand x-c does not divide f~ from the
right, then x-c' divides f from the right for some conjugate c' of c.
So again, there is a polynomial p such that f = p*(x-c'),
deg p < deg f [Editor: "deg p < f" in the original].  The result now
follows by induction on degree of f.

[ Editor: the poster of this sketch is not named in the collected
material. ]

Quaternionic analysis
---------------------

In article <6390@pucc.Princeton.EDU>, Eric Schissel writes:

> i have been told that the usual definition, when
> modified to take into account the non-commutativity of quaternion
> multiplication, leaves no non-constant functions differentiable.

This is mildly incorrect; functions of the form az + b, a,b \in H, are
differentiable; your friend must be thinking of the octonions.  I know
of no references to this, but the result is very easy to derive and
makes a great exercise.  Big hint: imitate the derivation of the
Cauchy-Riemann equations.

-- Chris Long, Mathematics Department, Rutgers University

In article <Oct.25.16.48.33.1988.23546@topaz.rutgers.edu>,
clong@topaz.rutgers.edu (Chris Long) writes:
> This is mildly incorrect; functions of the form az + b, a,b \in H,
> are differentiable; your friend must be thinking of the octonions.

This is only true if the difference quotients are defined by

                [f(x) - f(y)] / (x-y).

One could equally use

                (x-y) \ [f(x) - f(y)],

in which case the coefficient would have to be on the right.  If both
are used, the coefficient must be real.

In any case, there are no non-linear differentiable functions on the
quaternions.

-- Herman Rubin <hrubin@l.cc.purdue.edu>

Here are some references for the original message:

Sudbery, A. Quaternionic Analysis, Math. Proc. Camb. Phil. Soc. (1979),
85, 199-225 (cf. Theorem 1, p. 206).

Devours, C.A.  The Quaternionic Calculus, Amer. Math. Monthly 80 (1973)
995-1008.

R. Fueter (cited in the above) did much of the original work in the
1930's.

-- Paul Scofield (using feuerman@symcom.math.uiuc.edu's user id)

I remember that there were some attempts at analysis over Clifford
algebras.  [I am sorry but the only references for Clifford algebras I
know of are aimed at topologists.  Husemoller's `Fiber bundles' is
one.]  This was in a book called `Clifford analysis' (what else).  All
this is apparently relevant to `Superphysics'.  You may want to ask
mathematical physicsists for more references.

BTW, I ought to explain that the algebra of quaternions is a Clifford
algebra.

-- vkr@osupyr.mast.ohio-state.edu

> [I am sorry but the only references for Clifford algebras I know of
> are aimed at topologists. Husemoller's `Fiber bundles' is one.]

There is a better algebraic and geometric treatment in Artin's
_Geometric Algebra._  The material there should be comprehensible to
anyone with, say, enough geometric knowledge to understand GR.

At one point, Ernst Snapper of Dartmouth was working on a textbook on
Clifford Algebras (I recall reading proof for it in 1977 or so).  Does
anyone out there know if he ever published it?

-- Kevin

I missed the original query about quaternionic analysis and Clifford
algebras, but here are some more references on it:

Ian R. Porteous: "Topological Geometry"
Van Nostrand Reinhold 1969, ISBN 0 442 06605 8
still in print (paperback) from Cambridge University Press,
ISBN 0 521 29839 3

This fat book has lots of Truly Weird Stuff in it.  It was used as a
third year undergraduate text at Liverpool University; they must have
had one of the strangest curricula anywhere.  Cayley numbers, anyone?

Beltrametti and Cassinelli's "The Logic of Quantum Mechanics"
(Addison-Wesley, 1981; now from Cambridge University Press) has a
couple of pages on operators in quaternionic Hilbert spaces.  They
refer to two papers by G.C.Emch - Helv. Phys. Acta 36 (1963) pp. 739 &
770 - and an unpublished paper by Beltrametti.

Trivia: what does the quaternionic equivalent of the Mandelbrot set
look like?

-- Jack Campin <jack@cs.glasgow.ac.uk>

I've missed most of the discussion on quaternionic analysis so excuse
this posting if this repeats what has already been asked.

I'd be most interested in any references on bi-quaternions
particularly if relevant to the group of isometries of Euclidean
3-space.  In the early part of this century E. Study, a German
mathematician, wrote about a Hyperquadric in projective 7-space.  This
hyperquadric, sometimes called the Study Quadric, defines the group of
proper isometries of R^3.  The eight coordinates of a point on this
quadric can be split symmetrically to yield two quaternions, hence the
name bi-quaternion.  Apart from Study's work, written in German, and
two or three other minor references little else seems to have been
written.

Any References come to mind?

-- Ross McAree, Robotics Laboratory, University of Melbourne