Is e^x zero in some extended number system?

I came across the following a couple of days ago
and I'm curious if any related work has been done
along these lines.

Paul Dienes, "The exponential function in linear algebras",
Quarterly Journal of Mathematics (Oxford) 1 (1930), 300-309.
http://www.emis.de/cgi-bin/JFM-item?56.0151.02

The text of the first two paragraphs follows.

"The introduction of complex numbers was chiefly
suggested by the problem of determining the zeros
of polynomials. Some integral functions, such as
e^x, have no zero in the field of complex numbers.
This fact suggests the following question. Can we
generalize the idea of number to such an extent
that the exponential function may have a zero in
the extended field?"

"We shall prove in this Note that the exponential
function has no zero in the linear associative
algebra to a finite base, and that it has no
zero in finite non-associative linear algebras.
[Dienes assumes, of course, that the algebra
product has a multiplicative identity.] This
result extends to a large class of algebras to
an infinite base. In particular, the exponential
function has no zero in the tensor algebra of
relativity theory and it misses only singular
tensor values which do not divide some tensor.
Moreover, in Hilbert's [*] algebra of infinite
bounded matrices, so important in atom mechanics,
the exponential function has no absolutely
bounded matrix zero."

[*] "D. Hilbert, 'Grundzüge einer allgemeinen
Theorie der linearen Integralgleichungen',
Leipzig (1912), pp. 128-9."


Dave L. Renfro

.

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