Re: Number theory question


Rotwang wrote:
Given a prime p, is it always the case that the multiplicative group of non-zero integers mod p is isomorphic to the cyclic group of order p-1? This will be the case iff there exists a non-zero integer n s.t. n^x mod p !=1 for x=1,2,...p-2. I wrote a short program to test this, and assuming I didn't screw up, which I almost certainly did, it is true for all primes <10000. I have no idea how to go about proving it in general. Can anyone help?
****************************************************************************
Hi:
Perhaps even simpler and more general than Arturo's explanation is the
fact that any finite subgroup of the multiplicative group of any field
is always cyclic. Since
the ring Z/pZ , usually denoted by Z_p, is a field for p prime, you're
done.
Regards
Tonio

.

Relevant Pages

  • Re: Number theory question
    ... Rotwang wrote: ... fact that any finite subgroup of the multiplicative group of any field ... generators, and that n=sum_phi. ...
    (sci.math)
  • Re: Number theory question
    ... Perhaps even simpler and more general than Arturo's explanation is the ... fact that any finite subgroup of the multiplicative group of any field ... multiplicative group of classes relatively prime to n is cyclic) iff n ...
    (sci.math)
  • Re: algebra~~.
    ... > let me advice, please~ ... This is a particular case of the theorem "A finite subgroup of the ... Dubuque at a thread called "multiplicative group of residues mod p" at ... Jose Carlos Santos ...
    (sci.math)