Re: prime numbers and African artifact
Miguel Lerma (mlerma@albert.ma.utexas.edu)
13 Jul 1995 15:38:34 GMT
Alistair J. R. Young (avatar@arkane.demon.co.uk) wrote:
: In article <3trjtv$hal@news.iastate.edu>
: rhawkins@iastate.edu "Rick Hawkins" writes:
: > But only half- credit, since it's the wrong answer. 1 is not prime.
: > --
: > R E HAWKINS
: > rhawkins@iastate.edu
: Correct me if I'm wrong but if a prime number is only divisible by itself
: and 1, 1 is prime. What else is it divisible by?
That is not the right definition of a prime number. Number 1 is excluded
as "prime" in order to simplify the statement of the unique factorization
theorem. Hence, we can say that any natural number can be writen in a
unique way as a product of (positive) prime factors, without having to
add "different from 1".
In general, in any ring with identity the following classes of elements
can be defined:
1. Units: invertible elements (in Z the units are 1 and -1).
2. Irreducible: An element c is "irreducible" iff c is nonzero,
nonunit, and c = ab implies a or b is a unit.
3. Primes: An element p is prime iff p is nonzero, nonunit, and
if p divides ab then p divides a or p divides b.
Note that the classical definition of prime number in Z is actually
that of irreducible element. Actually in Z, prime elements and
irreducible elements are the same. However, there are rings where
they are different, for instance in the ring of numbers of the
form a + b sqrt(10) (a,b in Z), the numbers 2, 3, 4 + sqrt(10)
and 4 - sqrt(10) are irreducible, but not prime.
Finally, an unique factorization domain is a ring which is
an integral domain (conmutative and with no zero divisors)
such that every nonzero nonunit element can be writen as a
product of irreducible elements which is unique except for
the order of the factors and product by units. The unique
factorization theorem in Z can be stated by saying that Z is
a unique factorization domain.
Miguel A. Lerma
7/13/95
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