Re: zero-divisors

In article <e3eequ$eum$1@xxxxxxxxxxxxxxxxx>,
rusin@xxxxxxxxxxxxxxxxxxxxx (Dave Rusin) wrote:

In article <cjnelson9-5B337F.19111504052006@xxxxxxxxxxxxxxxx>, Clifford
Nelson <cjnelson9@xxxxxxxxxxx> wrote (amidst some incomprehensibilities):

The attention given to quaternions and not to four dimensional fields  
over the rational numbers, or over the integers congruence modulo a  
prime number, by so many people, suggests that there is some kind of  
dead end for rational numbers or integers congruence modulo a prime;  

It is in fact true that a "dead end" of sorts occurs in dimension 4
over the reals (that is, there are no associative division algebras
in higher dimensions) but there is no such "dead end" over Q nor
over F_p . In other words, CN has it exactly backwards.

This is especially humorous since the algebras described earlier in
his post contain or are related to the field Q[X]/(X^4+X^3+X^2+X+1)
of the fifth roots of unity, which is indeed a dimension-4 extension of
the rational numbers which has attracted a fair bit of attention.

dave

I wrote:

"The attention given to quaternions and not to four dimensional fields  
over the rational numbers, or over the integers congruence modulo a  
prime number, by so many people, suggests that there is some kind of  
dead end for rational numbers or integers congruence modulo a prime;  
that is, something you need irrational numbers for as coordinates.  
What can be done with exact rational numbers or integers congruence  
modulo a prime as coordinates and what can't be done?"

I was referring to a dead end in coordinate geometry.

Cliff Nelson

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