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The Importance of 1 Not Being a Prime Number

Date: 04/25/2009 at 02:31:53
From: Reg
Subject: Understanding fields

Does division by zero make sense in a field of cardinality = 1 ?

Everyone tells me you can't divide by zero.  Well, I figure since in
this case there is only one element, the multiplicative and additive
identities must be the same so 0=1.  That means we have 1/0 = 1/1 = 1.

Date: 04/25/2009 at 04:23:04
From: Doctor Jacques
Subject: Re: Understanding fields

Hi Reg,

The field axioms explicitly specify that 0 and 1 are distinct 
elements: there is no "trivial field" with only one element.

Note, however, that there is a trivial ring, with only one element
0 = 1.  The difference is a matter of convention.  In such cases, we
try to use the most convenient definition.  This is a delicate balance
between the following objectives:

* We make definitions as general as possible, to avoid excluding some
  cases that may prove interesting.

* We do not make definitions too general, to avoid having to consider
  a lot of special cases separately when we use them.

In this case, the trivial ring appears quite naturally in many
circumstances, and we do not want to exclude it, because that would 
produce many special cases that we want to avoid.

Why don't we accept a "trivial field" with one element?

One possible reason is linked to what you are describing, although you
should consider that in a slightly different perspective.  One of the
field axioms states that every non-zero element has an inverse.  If
there were no non-zero elements, this axiom would indeed be 
(vacuously) true, since there would be nothing to check.  However,
this field would be exactly the same as the trivial ring, and would
bring nothing new.  Another way to see this is to consider that the
axiom in question states that the non-zero elements of a field
constitute a group under multiplication; however, the empty set is not
a group (since it contains no identity element).

Another aspect is related to the fact that, in a finite field, the
number of elements is a power of a prime; this theorem is used in a
lot of places.  However, 1 is not considered a prime number (although
this was not always the case--mathematical definitions evolve with
time...); if you wonder why, have a look at:

  Why is 1 Not Considered Prime? 

  Wolfram Mathworld: Prime Number 

Another consideration is that a big part of linear algebra would break
down if we allowed a trivial field: in particular, any vector space
over that field would also contain only one element (0), and it would
be impossible to define the dimension of that vector space.  If T is
the "trivial field", the vector spaces T^k and T^m would both consist
of a single element, and would be isomorphic.  This is definitely
something we want to avoid.  By the way, this is probably the most
compelling reason not to consider 1 as a prime.

Please feel free to write back if you want to discuss this further.

- Doctor Jacques, The Math Forum 
Associated Topics:
College Modern Algebra
Middle School Prime Numbers

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